P A R A B O L A. a parabola an ellipse a hyperbola a recta ngular hyperbola e = 1 ; D 0 0 < e < 1 ; D 0 D 0 ; e > 1 ; e > 1 ; D 0

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1 J-Mathematics. CONIC SCTIONS : A conic section, or conic is the locus of a point which moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line. ( a ) The fixed point is called the focus. ( b ) The fixed straight line is called the directrix. ( c ) The constant ratio is called the eccentricity denoted by e. ( d ) The line passing through the focus & perpendicular to the directrix is called the axis. ( e ) A point of intersection of a conic with its axis is called a vertex.. GNR AL QUATION OF A CONIC : FOCAL DIRCTRIX PROPRTY : The general equation of a conic with focus (p, q) & directrix lx + my + n = 0 is : (l + m ) [(x p) + (y q) ] = e (lx + my + n) ax + hxy + by + gx + fy + c = 0. DISTINGUISHING BTWN TH CONIC : The nature of the conic section depends upon the position of the focus S w.r.t. the directrix & also upon the value of the eccentricity e. Two different cases arise. Case (i) W hen t he focus lie s on the directrix : In this case D abc + fgh af bg ch = 0 & the general equation of a conic represents a pair of straight lines and if : e > the lines will be real & distinct intersecting at S. e = the lines will coincident. e < the lines will be imaginary. Case (ii) W hen t he focus doe s not lie on the directrix : The conic represents: a parabola an ellipse a hyperbola a recta ngular hyperbola e = ; D 0 0 < e < ; D 0 D 0 ; e > ; e > ; D 0 h = ab h ab h > ab h > ab ; a + b = 0 4. PAR ABOLA : P A R A B O L A NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65 A parabola is the locus of a point which moves in a plane, such that its distance from a fixed point (focus) is equal to its perpendicular distance from a fixed straight line (directrix). Standard equation of a parabola is y = 4ax. For this parabola : (i) Vertex is (0, 0) (ii) Focus is (a, 0) (iii) Axis is y = 0 (iv) Directrix is x + a = 0 ( a ) Focal distance : The distance of a point on the parabola from the focus is called the focal distance of the point. ( b ) Focal chord : A chord of the parabola, which passes through the focus is called a focal chord. ( c ) Double ordi nate : A chord of the parabola perpendicular to the axis of the symmetry is called a double ordinate. ( d ) Latus rectum : A double ordinate passing through the focus or a focal chord perpendicular to the axis of parabola is called the latus rectum. For y = 4ax.

2 J-Mathematics Length of the latus rectum = 4a. Length of the semi latus rectum = a. nds of the latus rectum are L(a, a ) & L'(a, a ) Note that : (i) Perpendicular distance from focus on directrix = half the latus rectum. (ii) Vertex is middle point of the focus & the point of intersection of directrix & axis. (iii) Two parabolas are said to be equal if they have the same latus rectum. 5. PAR A MTRIC RPRSNTATION : The simplest & the best form of representing the co-ordinates of a point on the parabola is (at, at). The equation x = at & y = at together represents the parabola y = 4ax, t being the parameter. 6. TYP OF PAR ABOLA : Four standard forms of the parabola are y = 4ax ; y = 4ax ; x = 4ay ; x = 4ay X' Z M T A x= a Z' Y L P(x,y) (a,0) S N X Y' L' P' ( a,0) Y Z X' S 0 X Y' Z' X' Z' (0,a) 0 Y S y= a Y' X Z Z' X' Y y=a 0 S (0, a) y = 4ax y = 4ax x = 4ay x = 4ay Y' Z X \ P a r a b o l a Ver tex F o c u s A x i s Di rec tr i x Length of nds of P a r a m e t r i c F o c a l Latus rectum Latus rectum e q u a t i o n l e ng t h y = 4ax (0,0) (a,0) y = 0 x= a 4 a (a, ±a) (at,at) x + a y = 4ax (0,0) ( a,0) y = 0 x =a 4 a ( a, ±a) ( at,at) x a x = +4ay (0,0) (0,a) x = 0 y= a 4 a (±a, a) (at,at ) y + a x = 4ay (0,0) (0, a) x = 0 y = a 4 a (±a, a) (at, at ) y a (y k) = 4a(x h) (h,k) (h+a,k) y =k x+a h =0 4 a (h+a, k±a) (h+at,k+at) x h+a (x p) = 4b(y q) (p,q) (p, b+q) x = p y+b q=0 4b (p±a,q+a) (p+at,q+at ) y q+b Illustration : Find the vertex, axis, directrix, focus, latus rectum and the tangent at vertex for the parabola 9y 6x y 57 = 0. The given equation can be rewritten as Hence the vertex is 6, 6 The axis is y = 0 y = The directrix is X + A = 0 x + 0 x The focus is X = A and Y = 0 focus = 485, y x 9 6 which is of the form Y = 4AX. 6 4 x and 6 9 y 0 NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65

3 J-Mathematics Length of the latus rectum = 4A = 6 9 The tangent at the vertex is X = 0 6 x. A ns. 6 Illustration : The length of latus rectum of a parabola, whose focus is (, ) and directrix is the line x 4y + = 0 is - (A) 7 7 (B) 4 (C) 7 (D) 4 7 The length of latus rectum = perp. from focus to the directrix 4() 4 () (4) 7 Ans. (D) Illustration : Find the equation of the parabola whose focus is ( 6, 6) and vertex (, ). Let S( 6, 6) be the focus and A(, ) is vertex of the parabola. On SA take a point K(x, y ) such that SA = AK. Draw KM perpendicular on SK. Then KM is the directrix of the parabola. Since A 6 x 6 y bisects SK,,, 6 + x = 4 and 6 + y = 4 or (x, y ) = (, 0) Hence the equation of the directrix KM is y 0 = m(x )... (i) Also gradient of SK = y 0 = x 0 ( 6) ( 6) = 6 8 (from (i)) = ; m (x,y) P M S A K ( 6, 6) (,) (x,y ) x + y = 0 is the directrix Next, let PM be a perpendicular on the directrix KM from any point P(x, y) on the parabola. From NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65 Illustration 4 : SP = PM, the equation of the parabola is x 6 y 6 x y or 5(x + y + x + y + 7) = (x + y ) or 4x + y 4xy + 04x + 48y 4 = 0 or (x y) + 04x + 48y 4 = 0. A n s. The extreme points of the latus rectum of a parabola are (7, 5) and (7, ). Find the equation of the parabola. Focus of the parabola is the mid-point of the latus rectum. S is (7, 4). Also axis of the parabola is perpendicular to the latus rectum and passes through the focus. Its equation is 0 y 4 = x 7 y 4 5 Length of the latus rectum = (5 ) = Hence the vertex of the parabola is at a distance /4 = 0.5 from the focus. We have two parabolas, one concave rightwards and the other concave leftwards. The vertex of the first parabola is (6.5, 4) and its equation is (y 4) = (x 6.5) and it meets the x-axis at (4.5, 0). The equation of the second parabola is (y 4) = (x 7.5). It meets the x-axis at ( 0.5, 0). Ans.

4 J-Mathematics Do yourself - : ( i ) Name the conic represented by the equation ax by, where a, b R, a, b, > 0. (ii) Find the vertex, axis, focus, directrix, latus rectum of the parabola 4y + x 0y + 67 = 0. (iii) Find the equation of the parabola whose focus is (, ) and whose vertex is (, ). Also find its axis and latus rectum. ( i v ) Find the equation of the parabola whose latus rectum is 4 units, axis is the line x + 4y = 4 and the tangent at the vertex is the line 4x y + 7 = POSITION OF A POINT RL ATIV TO A PAR A BOL A : The point (x, y ) lies outside, on or inside the parabola y = 4ax according as the expression y 4ax is posit ive, zero or negat ive. Illustration 5 : Find the value of for which the point (, ) lies inside the parabola y = 4x. Point (, ) lies inside the parabola y = 4x y 4ax < 0 4( ) < < 0 ( ) < 0 A n s. 8. CHORD JOINING TWO POINTS : The equation of a chord of the parabola y = 4ax joining its two points P(t ) and Q(t ) is y(t + t ) = x + at t Note : (i) If PQ is focal chord then t t =. a (ii) xtremities of focal chord can be taken as (at, at) & t, a t Illustration 6 : Through the vertex O of a parabola y = 4x chords OP and OQ are drawn at right angles to one another. Show that for all position of P, PQ cuts the axis of the parabola at a fixed point. The given parabola is y = 4x... (i) Let P t, t, Q t, t Slope of OP = Since OP OQ, t and slope of OQ = t t 4 t t or t t = 4 The equation of PQ is y(t + t ) = (x + t t ) 4 y t t = (x 4) [from (ii)] 4 t... (ii) 4 (x 4) y t t = 0 L + L = 0 variable line PQ passes through a fixed point which is point of intersection of L = 0 & L = 0 i.e. (4, 0) A n s. NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65

5 J-Mathematics 9. LIN & A PARABOLA : ( a ) The line y = mx + c meets the parabola y = 4ax in two points real, coincident or imaginary according as a cm condition of tangency is, c = a m. Note : Line y = mx + c will be tangent to parabola x = 4ay if c = am. ( b ) Length of the chord intercepted by the parabola y = 4ax on the line y = mx + c is : 4 a ( m ) ( a m c). m Note : Length of the focal chord making an angle with the x - axis is 4a cosec. Illustration 7 : If the line y = x + intersect the parabola y = 4x at two distinct points then set of values of is - (A) (, ) (B) (, /) (C) (/, ) (D) none of these Putting value of y from the line in the parabola - (x + ) = 4x 9x + (6 4)x + = 0 Do yourself - : D > 0 line cuts the parabola at two distinct points 4( ) 4.9 > > 0 < / Hence, (, /) ( i ) Find the value of 'a' for which the point (a, a) lies inside the parabola y = 8x. ( i i ) The focal distance of a point on the parabola (x ) = 6(y 4) is 8. Find the co-ordinates. A ns.(b) (iii) Show that the focal chord of parabola y = 4ax makes an angle with x-axis is of length 4a cosec. ( i v ) Find the condition that the straight line ax + by + c = 0 touches the parabola y = 4kx. ( v ) Find the length of the chord of the parabola y = 8x, whose equation is x + y =. NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65 0. LNGTH OF SUBTANGNT & SUBNORMAL : PT and PG are the tangent and normal respectively at the point P to the parabola y = 4ax. Then TN = length of subtangent = twice the abscissa of the point P (Subtangent is always bisected by the vertex) NG = length of subnormal which is constant for all points on the parabola & equal to its semilatus rectum (a).. TANGNT TO TH PARABOLA y = 4ax : ( a ) Point form : quation of tangent to the given parabola at its point (x, y ) is yy = a (x + x ) ( b ) Slope form : quation of tangent to the given parabola whose slope is 'm', is a, m 0 m y = mx + a a Point of contact is, m m P(at, at) T N G 5

6 J-Mathematics ( c ) Parametric form : quation of tangent to the given parabola at its point P(t), is ty = x + at Note : Point of intersection of the tangents at the point t & t is [ at t, a(t + t )]. Illustration 8 : A tangent to the parabola y = 8x makes an angle of 45 with the straight line y = x + 5. Find its equation and its point of contact. Let the slope of the tangent be m tan45 = m m m ( m) m = or As we know that equation of tangent of slope m to the parabola y = 4ax is y = mx + a m and point a a of contact is, m m for m =, equation of tangent is y = x and point of contact is, for m =, equation of tangent is y = x + 4 and point of contact is (8, 8) A n s. Illustration 9 : Find the equation of the tangents to the parabola y = 9x which go through the point (4, 0). quation of tangent to parabola y = 9x is y = mx + 9 4m Since it passes through (4, 0) 0 = 4m + m =, m 6m 40 m + 9 = 0 equation of tangent's are y = x 9 4 & y = 9 x 4 A n s. Illustration 0 : Find the locus of the point P from which tangents are drawn to the parabola y = 4ax having slopes m and m such that - (i) m m (constant) (ii) = 0 (constant) where and are the inclinations of the tangents from positive x-axis. quation of tangent to y = 4ax is y = mx + a/m Let it passes through P(h, k) m h mk + a = 0 (i) m m (m + m ) m m = k a. h h locus of P(h, k) is y ax = x 6 NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65

7 J-Mathematics (ii) = 0 tan( ) = tan 0 m m m m tan 0 (m + m ) 4m m = tan 0 ( + m m ) k 4a a tan 0 h h h NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65 k 4ah = (h + a) tan 0 locus of P(h, k) is y 4ax = (x + a) tan 0 A n s. Do yourself - : (i ) Find the equation of the tangent to the parabola y = x, which passes through the point (, 5). Find also the co-ordinates of their points of contact. ( i i ) Find the equation of the tangents to the parabola y = 6x, which are parallel and perpendicular (iii) respectively to the line x y + 5 = 0. Find also the co-ordinates of their points of contact. Prove that the locus of the point of intersection of tangents to the parabola y = 4ax which meet at an angle is (x + a) tan = y 4ax.. NORMAL TO TH PARABOLA y = 4ax : ( a ) Point form : quation of normal to the given parabola at its point (x, y ) is y y y = a (x x ) ( b ) Slope form : quation of normal to the given parabola whose slope is 'm', is y = mx am am foot of the normal is (am, am) ( c ) Parametric form : quation of normal to the given parabola at its point P(t), is y + tx = at + at Note : (i) Point of intersection of normals at t & t is (a(t + t + t t + ), at t (t + t )). (ii) If the normal to the parabola y = 4ax at the point t, meets the parabola again at the point t, (iii) then t = t t. If the normals to the parabola y = 4ax at the points t & t intersect again on the parabola at the point t then t t = ; t = (t + t ) and the line joining t & t passes through a fixed point ( a, 0). (iv) If normal drawn to a parabola passes through a point P(h,k) then k = mh am am, i.e. am + m (a h) + k = 0. This gives m + m + m = 0 ; m m +m m +m m = a h ; m a m m = where m, m, & m are the slopes of the three concurrent normals : Algebraic sum of slopes of the three concurrent normals is zero. Algebraic sum of ordinates of the three co-normal points on the parabola is zero. Centroid of the formed by three co-normal points lies on the axis of parabola (x-axis). 7 k a

8 J-Mathematics Illustration : Prove that the normal chord to a parabola y = 4ax at the point whose ordinate is equal to abscissa subtends a right angle at the focus. Let the normal at P at, at meet the curve at Q at, at PQ is a normal chord. and t t...(i) t By given condition at at t = from equation (i), t = then P(4a, 4a) and Q(9a, 6a) but focus S(a, 0) Slope of SP = 4a 0 4a a and Slope of SQ = = 4a 4 a 6a 0 = 6a 9a a 8a 4 x' y A y' S P (at, at ) Q x (at, at ) 4 Slope of SP Slope of SQ 4 PSQ = / i.e. PQ subtends a right angle at the focus S. Illustration : If two normals drawn from any point to the parabola y = 4ax make angle and with the axis such that tan. tan =, then find the locus of this point. Let the point is (h, k). The equation of any normal to the parabola y = 4ax is y = mx am am passes through (h, k) k = mh am am am + m(a h) + k = 0...(i) m, m, m are roots of the equation, then m. m. m = k a but m m =, m = k a m is root of (i) k k a (a h) k 0 a a k = 4ah Thus locus is y = 4ax. A n s. Illustration : Three normals are drawn from the point (4, 7) to the curve y 6x 8y = 0. Find the coordinates of the feet of the normals. The given parabola is y 6x 8y = 0... (i) Let the co-ordinates of the feet of the normal from (4, 7) be P( ). Now the equation of the tangent at P( ) to parabola (i) is y 8(x + ) 4(y + ) = 0 or ( 4)y = 8x + 8a (ii) Its slope = 8 4 quation of the normal to parabola (i) at () is y = 4 (x ) 8 It passes through (4, 7) 8 NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65

9 J-Mathematics 7 = 4 6 (4 )... (iii) 8 4 Also () lies on parabola (i) i.e. 6 8 = 0... (iv) Do yourself - 4 : Putting the value of from (iii) in (iv), we get ( 4) 96 8( 4) = 0 ( ) = 0 ( 64) = 0 ( 6)( + 4) = 0 =0, 6, 4 from (iii), = 0 when = 0; = 8, when = 6 ; = when = 4 Hence the feet of the normals are (0, 0), (8, 6) and (, 4) Ans. ( i ) If three distinct and real normals can be drawn to y = 8x from the point (a, 0), then - (A) a > (B) a (, 4) (C) a > 4 (D) none of these ( i i ) Find the number of distinct normal that can be drawn from (, ) to the parabola y 4x y = 0. (iii) If x + y + k = 0 is a normal to the parabola y = 6x, then find the value of k. ( i v ) Three normals are drawn from the point (7, 4) to the parabola x 8x 6y = 0. Find the co-ordinates of the feet of the normals.. AN IMPORTANT CONCPT : If a family of straight lines can be represented by an equation P + Q + R = 0 where is a parameter and P, Q, R are linear functions of x and y then the family of lines will be tangent to the curve Q = 4PR. NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65 Illustration 4 : If the equation m (x + ) + m(y ) + = 0 represents a family of lines, where 'm' is parameter then find the equation of the curve to which these lines will always be tangents. m (x + ) + m(y ) + = 0 The equation of the curve to which above lines will always be tangents can be obtained by equating its discriminant to zero. (y ) 4(x + ) = 0 y 4y + 4 4x 4 = 0 y = 4(x + y) A n s. 4. PAIR OF TANGNTS : The equation of the pair of tangents which can be drawn from any point P(x, y ) outside the parabola to the parabola y = 4ax is given by : SS = T where : S y 4ax ; S y 4ax ; T yy a (x + x ). 5. DIRCTOR CIRCL : Locus of the point of intersection of the perpendicular tangents to the parabola y = 4ax is called the director circle. It s equation is x + a = 0 which is parabola s own directrix. Illustration 5 : The angle between the tangents drawn from a point ( a, a) to y = 4ax is - (A) /4 (B) / (C) / (D) /6 The given point ( a, a) lies on the directrix x = a of the parabola y = 4ax. Thus, the tangents are at right angle. Ans. (B) Illustration 6 : The circle drawn with variable chord x + ay 5 = 0 (a being a parameter) of the parabola y = 0x as diameter will always touch the line - (A) x + 5 = 0 (B) y + 5 = 0 (C) x + y + 5 = 0 (D) x y + 5 = 0 9

10 J-Mathematics Clearly x + ay 5 = 0 will always pass through the focus of y = 0x i.e. (5, 0). Thus the drawn circle will always touch the directrix of the parabola i.e. the line x + 5 = 0. Ans.(A) Do yourself - 5 : ( i ) If the equation x + y = 0 represents a family of lines, where '' is parameter, then find the equation of the curve to which these lines will always be tangents. ( i i ) Find the angle between the tangents drawn from the origin to the parabola, y = 4a(x a). 6. CHORD OF CONTACT : quation of the chord of contact of tangents drawn from a point P(x, y ) is yy = a(x + x ) Note : The area of the triangle formed by the tangents from the point (x, y ) & the chord of contact is y / 4 ax a / S i.e. a, also note that the chord of contact exists only if the point P is not inside. Illustration 7 : If the line x y = 0 intersect the parabola y = 8x at P & Q, then find the point of intersection of tangents at P & Q. Let (h, k) be point of intersection of tangents then chord of contact is yk = 4(x + h) 4x yk + 4h = 0 But given line is x y = 0 Comparing (i) and (ii) 4 k 4h... (i)... (ii) h =, k = 4 point (, 4) A n s. Illustration 8 : Find the locus of point whose chord of contact w.r.t. to the parabola y = 4bx is the tangent of the parabola y = 4ax. quation of tangent to y = 4ax is y = mx + 0 a m... (i) Let it is chord of contact for parabola y = 4bx w.r.t. the point P(h, k) quation of chord of contact is yk = b(x + h) y = b x bh k k From (i) & (ii) m = locus of P is y = b a bh 4b h, a k m k k 7. CHORD WITH A GIVN MIDDL POINT :... (ii) 4b x. A ns. a quation of the chord of the parabola y = 4ax whose middle point is (x, y ) is y y = This reduced to T = S, where T yy a (x + x ) & S y 4ax. a x x. y NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65

11 J-Mathematics Illustration 9 : Find the locus of middle point of the chord of the parabola y = 4ax which pass through a given (p, q). Let P(h, k) be the mid point of chord of the parabola y = 4ax, so equation of chord is yk a(x + h) = k 4ah. Since it passes through (p, q) qk a(p + h) = k 4ah Required locus is y ax qy + ap = 0. Illustration 0 : Find the locus of the middle point of a chord of a parabola y = 4ax which subtends a right angle at the vertex. The equation of the chord of the parabola whose middle point is ( ) is y a(x + ) = 4a Do yourself - 6 : y ax = a y ax or a... (i) Now, the equation of the pair of the lines OP and OQ joining the origin O i.e. the vertex to the points of intersection P and Q of the chord with the parabola y = 4ax is obtained by making the equation homogeneous by means of (i). Thus the equation of lines OP and OQ is 4ax y ax y = a y ( a) 4axy + 8a x = 0 If the lines OP and OQ are at right angles, then the coefficient of x + the coefficient of y = 0 Therefore, a + 8a = 0 = a( 4a) Hence the locus of ( ) is y = a(x 4a) ( i ) Find the equation of the chord of contacts of tangents drawn from a point (, ) to the parabola x = y. ( i i ) Find the co-ordinates of the middle point of the chord of the parabola y = 6x, the equation of which (iii) is x y + 8 = 0 Find the locus of the mid-point of the chords of the parabola y = 4ax such that tangent at the extremities of the chords are perpendicular. 8. DIAMTR : NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65 The locus of the middle points of a system of parallel chords of a Parabola is called a DIAMTR. quation to the diameter of a parabola is y = a/m, where m = slope of parallel chords. 9. IMPORTANT HIGHLIGHTS : ( a ) If the tangent & normal at any point P of the parabola intersect the axis at T & G then ST = SG = SP where S is the focus. In other words the tangent and the normal at a point P on the parabola are the bisectors of the angle between the focal radius SP & the perpendicular from P on the directrix. From this we conclude that all rays emanating from S will become parallel to the axis of the parabola after reflection. ( b ) The portion of a tangent to a parabola cut off between the directrix & the curve subtends a right angle at the focus. X' K ) T x= a Y Y' S (a,0) M Q P (h,k) G P S X

12 J-Mathematics ( c ) The tangents at the extremities of a focal chord intersect at right angles on the directrix, and a circle on any focal chord as diameter touches the directrix. Also a circle on any focal radii of a point P (at, at) as diameter touches the tangent at the vertex and intercepts a chord of length a t on a normal at the point P. ( d ) Any tangent to a parabola & the perpendicular on it from the focus meet on the tangent at the vertex. (e) Semi latus rectum of the parabola y = 4ax, is the harmonic mean between segments of any focal chord of the parabola is ; a = b c b c i.e.. b c a ( f ) If the tangents at P and Q meet in T, then : (i) TP and TQ subtend equal angles at the focus S. (ii) ST = SP. SQ & Q T S(a,0) P (iii) The triangles SPT and STQ are similar. ( g ) Tangents and Normals at the extremities of the latus rectum of a parabola L (a,a) y = 4ax constitute a square, their points of intersection being ( a, 0) N G & (a, 0). ( a,0) S (a,0) Note : (a, a) L' (i) The two tangents at the extremities of focal chord meet on the foot of the directrix. (ii) Figure LNL'G is square of side a ( h ) The circle circumscribing the triangle formed by any three tangents to a parabola passes through the focus. Do yourself - 7 : (i) (ii) The parabola y = 4x and x = 4y divide the square region bounded by the line x = 4, y = 4 and the co-ordinates axes. If S, S, S are respectively the areas of these parts numbered from top to bottom; then find S : S : S. Let P be the point (, 0) and Q a point on the parabola y = 8x, then find the locus of the mid point of PQ. Miscellaneous Illustrations : Illustration : The common tangent of the parabola y = 8ax and the circle x + y = a is - (A) y = x + a (B) x + y + a = 0 (C) x + y + a =0 (D) y = x + a Any tangent to parabola is y = mx + a m Solving with the circle B 4AC = 0 gives m = ± Tangent y = ± x ± a x + (mx + a m ) = a x ( + m ) + 4ax + 4a m a = 0 Ans. (C,D) Illustration : If the tangent to the parabola y = 4ax meets the axis in T and tangent at the vertex A in Y and the rectangle TAYG is completed, show that the locus of G is y + ax = 0. Let P(at, at) be any point on the parabola y = 4ax. Then tangent at P(at, at) is ty = x + at Since tangent meet the axis of parabola in T and tangent at the vertex in Y. NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65

13 J-Mathematics Co-ordinates of T and Y are ( at, 0) and (0, at) respectively. Let co-ordinates of G be (x, y ). Since TAYG is rectangle. Mid-points of diagonals TY and GA is same x' G T y Y A P (at, at) x and x 0 at 0 x at... (i) y 0 0 at y at... (ii) y' liminating t from (i) and (ii) then we get x = a a or y ax or y ax 0 The locus of G(x, y ) is y + ax = 0 Illustration : If P(, ) is one end of the focal chord PQ of the parabola y + 4x + 4y = 0, then the slope of the normal at Q is - (A) / (B) (C) / (D) The equation of the tangent at (, ) to the parabola y + 4x + 4y = 0 is y + (x ) + (y + ) = 0 or x + 4y = 0 x + y = 0 Since the tangent at one end of the focal chord is parallel to the normal at the other end, the slope of the normal at the other end of the focal chord is. Ans.(A) Illustration 4 : Prove that the two parabolas y = 4ax and y = 4c(x b) cannot have common normal, other than the axis unless b/(a c) >. Given parabolas y = 4ax and y = 4c(x b) have common normals. Then equation of normals in terms of slopes are y = mx am am and y = m(x b) cm cm respectively then normals must be identical, compare the co-efficients y NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65 am am mb cm cm m[(c a)m + (b + c a)] = 0, m 0 ( other than axis) and m = or or + a c b a c b, m c a c a b m c a b c a 0 b a c 0 b a c Illustration 5 : If r, r be the length of the perpendicular chords of the parabola y = 4ax drawn through the vertex, then show that r 4 / / / r 6a r r. Since chord are perpendicular, therefore if one makes an angle then the other will make an angle (90 ) with x-axis

14 J-Mathematics Let AP = r and AQ = r If PAX = then QAX = 90 Co-ordinates of P and Q are (r cos, r sin) and (r sin, r cos) respectively. Since P and Q lies on y = 4ax x' y A y' r r P(rcos, rsin ) 90 x Q (r sin, r cos ) r sin 4ar cos and r cos 4ar sin 4a cos 4a sin r = and r sin = cos 4 / 4 / 4a cos 4a sin 6a r r. sin cos sin cos 4 /... (i) / / / / 4a cos 4a sin 6a. r r 6a and = 6a. 4a = / cos sin / 6a. 4a sin cos 4 / sin cos 4 / / sin cos 4 / 4 / 6a cos cos 4 / = 6a. 4a = r 4 / r / cos sin sin cos 4 / 4 / {from (i)} Illustration 6 : The area of the triangle formed by three points on a parabola is twice the area of the triangle formed by the tangents at these points. Let the three points on the parabola be (at,at ),(at,at ) and (at,at ) The area of the triangle formed by these points = [ at (at at ) + at (at at ) + at (a at )] = a (t t )(t t )(t t ). The points of intersection of the tangents at these points are (at t, a(t + t )), (at t, a(t + t )) and (at t, a(t + t )) The area of the triangle formed by these three points at t (at at ) at t (at at ) at t (at at ) = = a (t t )(t t )(t t ) Hence = Illustration 7 : Prove that the orthocentre of any triangle formed by three tangents to a parabola lies on the directrix. Let the equations of the three tangents be t y x at...(i) t y x at...(ii) and t y x at...(iii) The point of intersection of (ii) and (iii) is found, by solving them, to be (at t, a(t + t )) NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65

15 J-Mathematics The equation of the straight line through this point & perpendicular to (i) is y a(t + t ) = t (x at t ) i.e. y + t x = a(t + t + t t t )...(iv) Similarly, the equation of the straight line through the point of intersection of (iii) and (i) & perpendicular to (ii) is y +t x = a(t + t + t t t )...(v) and the equation of the straight line through the point of intersection of (i) and (ii) & perpendicular to (iii) is y + t x = a(t + t + t t t )...(vi) The point which is common to the straight lines (iv), (v) and (vi) i.e. the orthocentre of the triangle, is easily seen to be the point whose coordinates are x = a, y = a(t + t + t + t t t ) and this point lies on the directrix. NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65 ANSWRS FOR DO YOURSLF : (i) Parabola (ii) 7 5 Vertex :,, Axis : y = 5, Focus : 7 5, 4, Directrix : x ; LR = 4 (iii) 4x + y 4xy + 8x + 46y 7 = 0; Axis : x y = ; LR = 4 5 unit (iv) (x + 4y 4) = 0(4x y + 7) : (i) 8 8,, 7 7 (ii) ( 7, 8), (9, 8) (iv) kb = ac ( v ) 8 : (i) 4 x y + = 0, (, 6); x y + 4 = 0,, 4 (ii) x y + = 0, (, 4) ; x + y + 6 = 0, (6, 6) 4 : (i) C (ii) (iii) 4 8 (iv) (0, 0), ( 4, ) and (6, 8) 5 : (i) (y + ) = 8(x ) (ii) / 6 : (i) x = y + (ii) (4, ) (iii) y = a(x a) 7 : (i) : : (ii) y 4x + = 0 5

16 J-Mathematics XRCIS - 0 CHCK YOUR GRASP SLCT TH CORRCT ALTRNATIV (ONLY ON CORRCT ANSWR). Latus rectum of the parabola whose focus is (, 4) and whose tangent at vertex has the equation x + y = 7 5 is - (A) 5 (B) 0 (C) 0 (D) 5. Directrix of a parabola is x + y =. If it's focus is origin, then latus rectum of the parabola is equal to - (A) units (B) units (C) units (D) 4 units. Which one of the following equations represents parametrically, parabolic profile? (A) x = cos t ; y = 4 sint (B) x = cost ; y = 4 cos t (C) x tan t ; y sec t (D) t t x sin t ; y sin cos 4. Let C be a circle and L a line on the same plane such that C and L do not intersect. Let P be a moving point such that the circle drawn with centre at P to touch L also touches C. Then the locus of P is - (A) a straight line parallel to L not intersecting C (B) a circle concentric with C (C) a parabola whose focus is centre of C and whose directrix is L. (D) a parabola whose focus is the centre of C and whose directrix is a straight line parallel to L. 5. If (t, t) is one end of a focal chord of the parabola y = 4x then the length of the focal chord will be- (A) t t (B) t t t t (C) t t t t (D) none 6. From the focus of the parabola y = 8x as centre, a circle is described so that a common chord of the curves is equidistant from the vertex and focus of the parabola. The equation of the circle is - (A) (x ) + y = (B) (x ) + y = 9 (C) (x + ) + y = 9 (D) x + y 4x = 0 7. The point of intersection of the curves whose parametric equations are x = t +, y = t and x = s, y = /s is given by - (A) (4, ) (B) (, ) (C) (, 4) (D) (, ) 8. If M is the foot of the perpendicular from a point P of a parabola y = 4ax to its directrix and SPM is an equilateral triangle, where S is the focus, then SP is equal to - (A) a (B) a (C) a (D) 4a 9. Through the vertex O of the parabola y = 4ax, variable chords OP and OQ are drawn at right angles. If the variable chord PQ intersects the axis of x at R, then distance OR : (A) varies with different positions of P and Q (C) equals latus rectum of the parabola 6 (B) equals the semi latus rectum of the parabola (D) equals double the latus rectum of the parabola 0. The triangle PQR of area A is inscribed in the parabola y = 4ax such that the vertex P lies at the vertex of the parabola and the base QR is a focal chord. The modulus of the difference of the ordinates of the points Q and R is - (A) A a (B) A a (C) A a (D) 4A a NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65

17 J-Mathematics. Point P lies on y = 4ax & N is foot of perpendicular from P on its axis. A straight line is drawn parallel to the axis to bisect NP and meets the curve in Q. NQ meets the tangent at the vertex in a point T such that AT = k NP, then the value of k is : (where A is the vertex) (A) / (B) / (C) (D) none. The tangents to the parabola x = y + c from origin are perpendicular then c is equal to - (A) (B) (C) (D) 4. The locus of a point such that two tangents drawn from it to the parabola y = 4ax are such that the slope of one is double the other is - (A) y ax 9 (B) y ax (C) y 4 9 9ax (D) x 4ay 4. T is a point on the tangent to a parabola y = 4ax at its point P. TL and TN are the perpendiculars on the focal radius SP and the directrix of the parabola respectively. Then - (A) SL = (TN) (B) (SL) = (TN) (C) SL = TN (D) (SL) = (TN) 5. The equation of the circle drawn with the focus of the parabola (x ) 8y = 0 as its centre and touching the parabola at its vertex is : (A) x + y 4 y = 0 (B) x + y 4 y + = 0 (C) x + y x 4 y = 0 (D) x + y x 4 y + = 0 6. Length of the normal chord of the parabola, y = 4x, which makes an angle of 4 with the axis of x is- (A) 8 (B) 8 (C) 4 (D) 4 7. Tangents are drawn from the point (, ) on the parabola y = 4x. The length, these tangents will intercept on the line x = : NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65 (A) 6 (B) 6 (C) 6 (D) none of these 8. Locus of the point of intersection of the perpendiculars tangent of the curve y + 4y 6x = 0 is : (A) x = 0 (B) x + = 0 (C) y + = 0 (D) x + 5 = 0 9. Tangents are drawn from the points on the line x y + = 0 to parabola y = 8x. Then the variable chords of contact pass through a fixed point whose coordinates are- (A) (, ) (B) (, 4) (C) (, 4) (D) (4, ) 0. The line 4x 7y + 0 = 0 intersects the parabola, y = 4x at the points A & B. The co-ordinates of the point of intersection of the tangents drawn at the points A & B are : (A) 7 5, (B) 5 7, (C) 5 7, (D) 7 5,. From the point (4, 6) a pair of tangent lines are drawn to the parabola, y = 8x. The area of the triangle formed by these pair of tangent lines & the chord of contact of the point (4, 6) is (A) (B) 4 (C) 8 (D) none. TP & TQ are tangents to the parabola, y =4ax at P & Q. If the chord PQ passes through the fixed point (-a, b) then the locus of T is - (A) ay = b (x b) (B) bx = a (y a) (C) by = a (x a) (D) ax = b (y b) 7

18 J-Mathematics. If the tangent at the point P (x,y ) to the parabola y = 4ax meets the parabola y = 4a (x + b) at Q & R, then the mid point of QR is - (A) (x + b, y + b) (B) (x b, y b) (C) (x, y ) (D) (x + b, y ) 4. Let PSQ be the focal chord of the parabola, y = 8x. If the length of SP=6 then, l(sq) is equal to(where S is the focus) - (A) (B) 4 (C) 6 (D) none 5. Two parabolas y = 4a(x l ) and x = 4a(y l ) always touch one another, the quantities l and l are both variable. Locus of their point of contact has the equation - (A) xy = a (B) xy = a (C) xy = 4a (D) none SLCT TH CORRCT ALTRNATIVS (ON OR MOR THAN ON CORRCT ANSWRS) 6. quation x x y + 5 = 0 represents - (A) a parabola with vertex (, ) (B) a parabola with vertex (, ) (C) a parabola with directrix y (D) a parabola with directrix y 5 7. The normals to the parabola y = 4ax from the point (5a, a) are - (A) y = x + a (B) x = y + a (C) y = x a (D) y = x + a 8. The equation of the lines joining the vertex of the parabola y = 6x to the points on it whose abscissa is 4, is - (A) y + x + = 0 (B) y x + =0 (C) x + y = 0 (D) x y = 0 9. The equation of the tangent to the parabola y = 9x which passes through the point (4, 0) is - (A) x + 4y + = 0 (B) x 4y + 6 = 0 (C) 9x 4y + 4 = 0 (D) 9x + 4y + 4 = 0 0. Consider the equation of a parabola y = 4ax, (a < 0) which of the following is false - (A) tangent at the vertex is x = 0 (B) directrix of the parabola is x = 0 (C) vertex of the parabola is at the origin (D) focus of the parabola is at ( a, 0) CHCK YOUR GRASP ANSWR KY X R CI S - Que A n s. C C B D A B B D C C B D A C D Q u e A n s. B B D C C A C C A C A, C C, D C, D B, C B, D 8 NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65

19 NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65 9 J-Mathematics XRCIS - 0 BRAIN TASRS SLCT TH CORRCT ALTRNATIVS (ON OR MOR THAN ON CORRCT ANSWRS). The straight line joining any point P on the parabola y = 4ax to the vertex and perpendicular from the focus to the tangent at P, intersect at R, then the equation of the locus of R is - (A) x + y ax = 0 (B) x + y ax = 0 (C) x + y ay = 0 (D) x + y ay = 0. Let A be the vertex and L the length of the latus rectum of parabola, y y 4x 7 = 0. The equation of the parabola with point A as vertex, L as the length of the latus rectum and the axis at right angles to that of the given curve is - (A) x + 4x + 8y 4 = 0 (B) x + 4x 8y + = 0 (C) x + 4x + 8y + = 0 (D) x + 8x 4y + 8 = 0. The parametric coordinates of any point on the parabola y = 4ax can be - (A) (at, at) (B) (at, at) (C) (asin t, asint) (D) (asint, acost) 4. PQ is a normal chord of the parabola y = 4ax at P, A being the vertex of the parabola. Through P a line is drawn parallel to AQ meeting the x-axis in R. Then the length of of AR is - (A) equal to the length of the latus rectum (B) equal to the focal distance of the point P. (C) equal to twice the focal distance of the point P. (D) equal to the distance of the point P from the directrix 5. The length of the chord of the parabola y = x which is bisected at the point (, ) is- (A) 5 (B) 4 5 (C) 4 50 (D) 5 6. If the tangents and normals at the extremities of a focal chord of a parabola intersect at (x, y ) and (x, y ) respectively, then - (A) x = x (B) x = y (C) y = y (D) x = y 7. Locus of the intersection of the tangents at the ends of the normal chords of the parabola y = 4ax is - (A) (a + x)y + 4a = 0 (B) (x + a)y + 4a = 0 (C) (y + a)x + 4a = 0 (D) none 8. The locus of the mid point of the focal radii of a variable point moving on the parabola, y = 4ax is a parabola whose (A) latus rectum is half the latus rectum of the original parabola (B) vertex is (a/, 0) (C) directrix is y-axis (D) focus has the co-ordinates (a, 0) 9. The equation of a straight line passing through the point (, 6) and cutting the curve y = x orthogonally is - (A) 4x + y 8 = 0 (B) x + y 9 = 0 (C) 4x y 6 = 0 (D) none 0. The tangent and normal at P (t), for all real positive t, to the parabola y = 4ax meet the axis of the parabola in T and G respectively, then the angle at which the tangent at P to the parabola is inclined to the tangent at P to the circle through the points P, T and G is - t (A) cot t (B) cot t (C) tan t (D) sin t

20 J-Mathematics. A variable circle is described to passes through the point (, 0) and tangent to the curve y = tan(tan x). The locus of the centre of the circle is a parabola whose - (A) length of the latus rectum is (B) axis of symmetry has the equation x + y = (C) vertex has the co-ordinates (/4, /4) 0 (D) none of these. AB, AC are tangents to a parabola y = 4ax. p p and p are the lengths of the perpendiculars from A, B and C respectively on any tangent to the curve, then p, p, p are in- (A) A.P. (B) G.P. (C) H.P. (D) none of these. Through the vertex O of the parabola, y = 4ax two chords OP and OQ are drawn and the circles on OP and OQ as diameter intersect in R. If, and are the angles made with the axis by the tangent at P and Q on the parabola and by OR then the value of cot + cot = (A) tan (B) tan( ) (C) 0 (D) cot 4. Two parabolas have the same focus. If their directrices are the x-axis & the y-axis respectively, then the slope of their common chord is - (A) (B) (C) 4/ (D) /4 5. Tangent to the parabola y = 4ax at point P meets the tangent at vertex A, at point B and the axis of parabola at T. Q is any point on this tangent and N is the foot of perpendicular from Q on SP, where S is focus. M is the foot of perpendicular from Q on the directrix then - (A) B bisects PT (B) B trisects PT (C) QM = SN (D) QM = SN 6. If the distance between a tangent to the parabola y = 4 x and a parallel normal to the same parabola is, then possible values of gradient of either of them are - (A) (B) + (C) 5 (D) If two distinct chords of a parabola x = 4ay passing through (a, a) are bisected on the line x + y =, then length of latus rectum can be - (A) (B) (C) 4 (D) 5 8. If PQ is a chord of parabola x = 4y which subtends right angle at vertex. Then locus of centroid of triangle PSQ (S is focus) is a parabola whose - (A) vertex is (0, ) (B) length of LR is 4/ (C) axis is x = 0 (D) tangent at the vertex is x = 9. Identify the correct statement(s) - (A) In a parabola vertex is the mid point of focus and foot of directrix. (B) P at, at & perpendicular. Q at, at are two points on y = 4ax such that t t =, then normals at P and Q are (C) There doesn't exist any tangent of y = 4ax which is parallel to x-axis. (D) At most two normals can be drawn to a parabola from any point on its plane. 0. For parabola y = 4ax consider three points A, B, C lying on it. If the centroid of ABC is (h, k ) & centroid of triangle formed by the point of intersection of tangents at A, B, C has coordinates (h, k ), then which of the following is always true - (A) k = k (B) k = k (C) k h h 4a (D) k h h BRAIN TASRS ANSWR KY X R CI S - Que A n s. B A, B A, B C D C A A,B,C,D A C, D Q u e A n s. B, C B A A, B A, C A,B,C,D A, B A,B,C A,B,C B, C 4a NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65

21 J-Mathematics XRCIS - 0 MISCLLANOUS TYP QUSTIONS MATCH TH COLUMN Following questions contains statements given in two columns, which have to be matched. The statements in Column-I are labelled as A, B, C and D while the statements in Column-II are labelled as p, q, r and s. Any given statement in Column-I can have correct matching with ON statement in Column-II.. Column-I Column- II (A) The normal chord at a point t on the parabola y = 4x (p) 4 subtends a right angle at the vertex, then t is (B) The area of the triangle inscribed in the curve y = 4x. (q) If the parameter of vertices are, and 4 is (C) The number of distinct normal possible from, to the (r) 4 4 parabola y = 4x is (D) The normal at (a, a) on y = 4ax meets the curve again (s) 6 at (at, at), then the value of t is NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65. Column-I Column- II (A) Area of a triangle formed by the tangents drawn from a (p) 8 point (, ) to the parabola y = 4(x + y) and their corresponding chord of contact is (B) Length of the latus rectum of the conic (q) 4 5{(x ) + (y ) } = (x + 4y 6) is (C) If focal distance of a point on the parabola y = x 4 is 5/4 (r) 4 and points are of the form (± a, b) then value of a + b is (D) Length of side of an equilateral triangle inscribed (s) 4/ 5 in a parabola y x y = 0 whose one angular point is vertex of the parabola, is ASSRTION & RASON These questions contain, Statement-I (assertion) and Statement-II (reason). (A) Statement-I is true, Statement-II is true ; Statement-II is correct explanation for Statement-I. (B) Statement-I is true, Statement-II is true ; Statement-II is NOT a correct explanation for Statement-I. (C) Statement-I is true, Statement-II is false. (D) Statement-I is false, Statement-II is true.. Statement-I : If normal at the ends of double ordinate x = 4 of parabola y = 4x meet the curve again at P and P' respectively, then PP' = unit. B e c a u s e Statement-II : If normal at t of y = 4ax meets the parabola again at t, then t = + t t. (A) A (B) B (C) C (D) D. Statement-I : The lines from the vertex to the two extremities of a focal chord of the parabola y = 4ax are at an angle of. B e c a u s e Statement-II : If extremities of focal chord of parabola are ( at, at ) and ( at, at ), then t t = (A) A (B) B (C) C (D) D

22 J-Mathematics. Statement-I : If P Q and P Q are two focal chords of the parabola y = 4ax, then the locus of point of intersection of chords P P and Q Q is directrix of the parabola. Here P P and Q Q are not parallel. B e c a u s e Statement-II : The locus of point of intersection of perpendicular tangents of parabola is directrix of parabola. (A) A (B) B (C) C (D) D COMPRHNSION BASD QUSTIONS Comprehension # Observe the following facts for a parabola : (i) (ii) Axis of the parabola is the only line which can be the perpendicular bisector of the two chords of the parabola. If AB and CD are two parallel chords of the parabola and the normals at A and B intersect at P and the normals at C and D intersect at Q, then PQ is a normal to the parabola. Let a parabola is passing through (0, ), (, ), (, ) & (, ) On t he basis of above i nformat ion, a nswer t he fol low i ng que st ions :. The vertex of the parabola is - (A), (B),. The directrix of the parabola is - (C) (, ) (D) (, ) (A) y 4 = 0 (B) y + = 0 (C) y + 4 = 0 (D) y 0. For the parabola y = 4x, AB and CD are any two parallel chords having slope. C is a circle passing through O, A and B and C is a circle passing through O, C and D, where O is origin. C and C intersect at - (A) (4, 4) (B) ( 4, 4) (C) (4, 4) (D) ( 4, 4) Comprehension # : If a source of light is placed at the fixed point of a parabola and if the parabola is a reflecting surface, then the ray will bounce back in a line parallel to the axis of the parabola. On t he basis of above i nformat ion, a nswer t he fol low i ng que st ions :. A ray of light is coming along the line y = from the positive direction of x-axis and strikes a concave mirror whose intersection with the xy-plane is a parabola y = 8x, then the equation of the reflected ray is - (A) x + 5y = 4 (B) x + y = 6 (C) 4x + y = 8 (D) 5x + 4y = 0. A ray of light moving parallel to the x-axis gets reflected from a parabolic mirror whose equation is y + 0y 4x + 7 = 0 After reflection, the ray must pass through the point - (A) (, 5) (B) (, 5) (C) (, 5) (D) ( 4, 5). Two ray of light coming along the lines y = and y = from the positive direction of x-axis and strikes a concave mirror whose intersection with the xy-plane is a parabola y = x at A and B respectively. The reflected rays pass through a fixed point C, then the area of the triangle ABC is - (A) 8 sq. unit (B) 9 sq. unit (C) 7 sq. unit (D) 5 sq. unit MISCLLANOUS TYP QUSTION ANSWR KY XRCIS - Match the Column. (A) (q); (B) (s); (C) (q); (D) (p). (A) (r); (B) (s); (C) (p); (D) (q) Assertion & Reason. C. D. B Comprehension Based Que st ions Comprehension # :. A. C. A Comprehension # :. C. B. A NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65

23 J-Mathematics XRCIS - 04 [A] CONCPTUAL SUBJCTIV XRCIS. Find the equation of parabola, whose focus is (, 0) and directrix is x + 5 = 0.. Find the vertex, axis, focus, directrix, latus rectum of the parabola x + y x + 5 = 0. Find the equation of the parabola whose focus is (, ) and whose vertex is (, ). Also find its axis and latus rectum. 4. If the end points P(t ) and Q( t ) of a chord of a parabola y = 4ax satisfy the relation t t = k (constant) then prove that the chord always passes through a fixed point. Find that point also? 5. Find the locus of the middle points of all chords of the parabola y = 4ax which are drawn through the vertex. 6. O is the vertex of the parabola y² = 4ax & L is the upper end of the latus rectum. If LH is drawn perpendicular to OL meeting OX in H, prove that the length of the double ordinate through H is 4a Find the length of the side of an equilateral triangle inscribed in the parabola, y = 4x so that one of its angular point is at the vertex. 8. Two perpendicular chords are drawn from the origin 'O' to the parabola y = x, which meet the parabola at P and Q. Rectangle POQR is completed. Find the locus of vertex R. 9. Find the set of values of in the interval [/, /], for which the point (sin, cos) does not lie outside the parabola y + x = Find the length of the focal chord of the parabola y = 4ax whose distance from the vertex is p.. If 'm' varies then find the range of c for which the line y = mx + c touches the parabola y = 8(x + ).. Find the equations of the tangents to the parabola y² = 6x, which are parallel & perpendicular respectively to the line x y + 5 = 0. Find also the coordinates of their points of contact.. Find the equations of the tangents of the parabola y² = x, which passes through the point (, 5). 4. Prove that the locus of the middle points of all tangents drawn from points on the directrix to the parabola y = 4ax is y²(x + a) = a(x + a)². NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65 5. Two tangents to the parabola y² = 8x meet the tangent at its vertex in the points P & Q. If PQ = 4 units, prove that the locus of the point of the intersection of the two tangents is y² = 8 (x + ). 6. Find the equation of the circle which passes through the focus of the parabola x = 4y & touches it at the point (6, 9). 7. In the parabola y² = 4ax, the tangent at the point P, whose abscissa is equal to the latus rectum meets the axis in T & the normal at P cuts the parabola again in Q. Prove that PT : PQ = 4 : Show that the normals at the points (4a, 4a) & at the upper end of the latus rectum of the parabola y² = 4ax intersect on the same parabola. 9. Show that the locus of a poi nt, such that t wo of the three normals draw n from it to the parabola y² = 4ax are perpendicular is y² = a(x a). 0. If the normal at P(8, ) to the parabola y² = 8x cuts it again at Q, then show that 9PQ = Prove that the locus of the middle point of portion of a normal to y² = 4ax intercepted between the curve & the axis is another parabola. Find the vertex & the latus rectum of the second parabola.

24 J-Mathematics. A variable chord PQ of the parabola y² = 4x is drawn parallel to the line y = x. If the parameters of the points P & Q on the parabola are p & q respectively, show that p + q =. Also show that the locus of the point of intersection of the normals at P & Q is x y =.. P & Q are the points of contact of the tangents drawn from the point T to the parabola y² = 4ax. If PQ be the normal to the parabola at P, prove that TP is bisected by the directrix. 4. The normal at a point P to the parabola y² = 4ax meets its axis at G. Q is another point on the parabola such that QG is perpendicular to the axis of the parabola. Prove that QG² PG² = constant. 5. Three normals to y² = 4x pass through the point (5, ). Show that one of the normals is given by y = x & find the equations of the others. CONCPTUAL SUBJCTIV XRCIS ANSWR KY X R C IS - 4 ( A ). y = 4(x + 4). Vertex, 8, focus 5, 8, axis : x =, directrix : y = 7, latus rectum = 8. (x y ) = 0(x + y 4), axis : x y = 0. latus rectum = ( ak, 0) 5. y = ax x = y 9. [ /, 5 / 6] [, / ] 0. 4a p 4. (, 4] [4, ). x y + = 0, (, 4) ; x + y + 6 = 0, (6, 6). x y + 4 = 0 ; x y + = 0 6. x + y + 8 x 8 y + 7 = 0. (a, 0) ; a 5. y = 4x + 7, y = x NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65

25 J-Mathematics XRCIS - 04 [B] BRAIN STORMING SUBJCTIV XRCIS. If from the vertex of a parabola a pair of chords be drawn at right angles to one another, & with these chords as adjacent sides a rectangle be constructed, then find the locus of the outer corner of the rectangle.. Two perpendicular straight lines through the focus of the parabola y² = 4ax meet its directrix in T & T respectively. Show that the tangents to the parabola parallel to the perpendicular lines intersect in the mid point of T T '.. Find the condition on a & b so that the two tangents drawn to the parabola y² = 4ax from a point are normals to the parabola x² = 4by. 4. TP & TQ are tangents to the parabola and the normals at P & Q meet at a point R on the curve. Prove that the centre of the circle circumscribing the triangle TPQ lies on the parabola y² = a(x a). 5. Let S is the focus of the parabola y = 4ax and X the foot of the directrix, PP' is a double ordinate of the curve and PX meets the curve again in Q. Prove that P'Q passes through focus. 6. Prove that on the axis of any parabola y² = 4ax there is a certain point K which has the property that, if a chord PQ of the parabola be drawn through it, then P K Q K also the coordinates of the point K. is same for all positions of the chord. Find 7. If (x, y ), (x, y ) and (x, y ) be three points on the parabola y = 4ax and the normals at these points meet in a point, then prove that x x y + x x y + x x y = 0 8. A variable chord joining points P(t ) and Q(t ) of the parabola y² = 4ax subtends a right angle at a fixed point t 0 of the curve. Show that it passes through a fixed point. Also find the co-ordinates of the fixed point. 9. Show that a circle circumscribing the triangle formed by three co-normal points passes through the vertex of the parabola and its equation is, (x +y ) (h+a) x ky = 0, where (h, k) is the point from where three concurrent normals are drawn. NOD6\\Data\04\Kota\J-Advanced\SMP\Maths\Unit#09\ng\0 PARABOLA.p65 0. A ray of light is coming along the line y = b from the positive direction of x-axis & strikes a concave mirror whose intersection with the xy-plane is a parabola y = 4 ax. Find the equation of the reflected ray & show that it passes through the focus of the parabola. Both a & b are positive. [R 95] BRAIN STORMING SUBJCTIV XRCIS ANSWR KY X R C I S - 4 ( B ). y² = 4a(x 8a). a² > 8b² 6. (a, 0) 8. [a(t² 0 + 4), at 0 ] 0. 4abx + (4a² b²)y 4a²b = 0 5