PARABOLA SYNOPSIS 1.S is the focus and the line l is the directrix. If a variable point P is such that SP

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PARABOLA SYNOPSIS.S is the focus and the line l is the directrix. If a variable point P is such that SP PM = where PM is perpendicular to the directrix, then the locus of P is a parabola.

1 PARABOLA SYNOPSIS.S is the focus and the line l is the directrix. If a variable point P is such that SP PM = where PM is perpendicular to the directrix, then the locus of P is a parabola... S ax + hxy + by + gx + fy + c = 0 represents (i) A circle if 0 a = b, h = 0 (ii) A parabola if 0 h = ab and af bg (iii) An ellipse if 0 h < ab (iv) An hyperbola if 0 h > ab i) If the axis of a parabola is parallel to x-axis, equation of the parabola will be of the form (y β) = 4a(x α) (or) (y β) = 4a(x α) (or) x = ay + by + c. ii) If the axis of the parabola is parallel to y-axis, equation of the parabola will be of the form (x α) = 4a(y β) (or) (x α) = 4a(y β) (or) y = ax + bx + c. 5. In the equation of the parabola (y β) = 4a(x α). i) Vertex = (α, β). ii) Focus = (α + a, β) iii) Ends of latustrectum = (α + a, β ± β a)

2 iv) Equation of axis is y = β v) Equation of directrix is x = α a. vi) Equation of latustrectum is x = α + a vii) Length of latustrectum = 4a. 6. In the equation of the parabola (x α) = 4a(y β) i) Vertex = (α, β) ii) Focus = (α, β + a) iii) Ends of latustrectum = (α± a, β + a) iv) Equation of axis is x = α v) Equation of directrix is y = β a vi) Equation of latustrectum is y = β + a vii) Length of latustrectum = 4a. 7. The focal distance of the point P(x, y ) on the parabola i) y = 4ax is SP = x + a ii) x = 4ay is SP = y + a. 8. The equation of the tangent at (x,y ) to the parabola y =4ax is yy =a(x+x ) i.e. S = 0. Slope of this tangent is a y 9. The line y = mx + c is a tangent to y = 4ax if c = a m. (i) The line y = mx + c is a tangent to x = 4ay if c = am. (ii). The condition for x = my + c is a normal to x = 4ay is c = -am - am 3 0. If m is the slope of any tangent to y = 4ax, then its equation is y = mx + a m and its point of a contact is m a,. m. The line lx + my + n = 0 is a tangent to the parabola y = 4ax if am = ln and n am point of contact =, l l. The line lx + my + n = 0 is a tangent to the parabola x = 4ay, if al = mn. and point of contact = l a, m n m 3. The line y = mx + c, touches the parabola y = 4a(x + a), if c = am + a/m

3 4. i) The line y = mx + c doesnot intersect the parabola y = 4ax (a > 0) then mc > a ii) The line y = mx+c intersects the parabola y = 4ax (a > 0) the mc < a. 5. The slopes of the tangents to the parabola y =4ax passing through (x, y ) are given by the equation m x - my + a = 0 6. Sum of the slopes = y /x Product of the slopes = a/x 7. Angle between the tangents drawn to the parabola y = 4ax from (x, y ) is α = tan - S x + a 8. The equation of the locus of the point of intersection of tangents to the parabola y =4ax which include an angle α is (y - 4ax) cot α=(x+a). 9. The locus of the points of intersection of perpendicular tangents to a parabola is the directrix of the parabola. 0. The tangents at the ends of a focal chord of the parabola meet on the directrix at right angles.. If the tangents are drawn from any point on the latusrectum to the parabola, they make complementary angles with the axis of the parabola.. a. The equation of the tangent at (at, at) is yt = x + at b. Slope of the tangent at (at, at) = t 3. Point of intersection of the tangents at t, t is (at t, at + t ) 4. If the tangents at P and Q on a parabola meet in T and if S is the focus then ST = SP.SQ 5. If m is the slope of any normal to y = 4ax then its equation is y = mx - am - am 3 and foot of the normal is (am, -am) 6. The equation of the normal at (x, y ) is y - y = - y a (x - x ) 7. a. The equation of the normal at (at, at) to y = 4ax is xt + y = at + at 3. b. Slope of the normal at (at, at) = -t 8. The line lx + my + n = 0 is a normal to the parabola y = 4ax if al 3 + alm + m n = 0 9. If the normals at (, y ), ( x, ) x + yy a x = x on the parabola y = 4ax meet again on the parabola, then y

4 30. From any point, three normals can be drawn to a parabola and the sum of the ordinates of the normals is zero. 3. The point of intersection of normals drawn at t, t to the parabola y = 4ax is [a (+t + t + t t ), - at t (t + t )] 3. If the normal at ( at, at ) cuts the curve again at (, ) at at, then t = t t 33. If the normals at the points t, t meet on the curve again at t 3, then (i) t t = (ii) t +t = -t If the normals at P, Q on the parabola meet on the curve at R, the centroid of the triangle PQR lies on the axis of the parabola. 35. From any point three normals can be drawn to a parabola and (i) At least one of the 3 normals is real (ii) The sum of their slopes is '0' (iii) The sum of the ordinates of their feet is '0' 36. The foot of the perpendicular from the focus on any tangent to a parabola lies on the tangent at the vertex. 37. The tangent at one end of a focal chord is parallel to the normal at the other end. 38. The orthocentre of the triangle formed by any three tangents to the parabola lies on the directrix of the parabola. 39. The orthocentre of the triangle formed by tangents at t, t, t 3 to the parabola y =4ax is (-a, a(t + t + t 3 + t t t 3 )) 40. The circum circle of the triangle formed by any three tangents to a parabola passes through its focus. 4. The length of the chord of contact of tangents drawn from (x, y ) to the parabola y = 4ax is y 4ax y + 4a a and area of the triangle formed by the tangents from (x, y ) to the ( ) parabola y y 4ax = 4ax and its chord of contact is 4. (a) Length of the chord cut-off by the parabola y = 4ax on the line y = mx + c is 4 m a amc + m, a 3 /.

5 (b) Length of the chord cut off by any conic on the line y = mx+c is x x and x are the roots of quadratic obtained by eliminating y. + m where x 43. The area of the triangle inscribed in the parabola y = 4ax is (y - y ) 8a (y - y 3 ) (y 3 - y ) where y, y, y 3 are the ordinates of the vertices. 44. The ends of focal chord of the parabola y = 4ax are ( x, y ), ( x, ) x x + yy = 3a y. Then 45. The equation of the chord of the parabola y =4ax joining the points ( at, at ),( at, at ) is y(t + t ) = x + at t. 46. If the points t, t are ends of a focal chord, then t t = -and ends of focal chord are (at a a, at),,. t t 47. The length of the chord joining the points t, t is a t - t ( t t ) a) Least length of focal chord is 4a b) Length of focal chord drawn at t is a(t + /t) 49. If a focal chord of the parabola y = 4ax makes an angle θ with the x-axis; then the length of the focal chord is 4acosec θ. 50. a) The length of the normal chord drawn at t is given by 4a( + t ) 3/ /t. b) The least length of normal chord is 6 3 a, when drawn at (a, a) 5. The length of the chord of the parabola y = 4ax passing through the vertex and making an angle 'θ' with the axes is 4acosθcosec θ. 5. If the normal at 't' on the parabola y = 4ax subtends a right angle (a) at its focus then t = ± (b) at its vertex then t = ±. 53. The equation of the chord of contact of tangents from ( x, y ) to the parabola is S = The equation of the chord of the parabola y = 4ax having its midpoint at ( x, y ) is S =S. 55. The semi lotus rectum of any conic is H.M. between the segments of a focal chord. 56. The circle described on any focal radius of a parabola as diameter touches tangent at the vertex. 57. The circle described on any focal chord of a parabola as diameter touches the directrix.

6 58. The equation of the common tangent to two parabolas y = 4ax and x = 4by is a /3 x + b /3 y + a /3 b /3 = 0

Drill Exercise - 1. Drill Exercise - 2. Drill Exercise - 3

Drill Exercise - 1. Drill Exercise - 2. Drill Exercise - 3 Drill Exercise - 1 1. Find the distance between the pair of points, (a sin, b cos ) and ( a cos, b sin ). 2. Prove that the points (2a, 4a) (2a, 6a) and (2a + 3 a, 5a) are the vertices of an equilateral