Maths Summer Vacation Assignment Package Solution

Transcription

1 Maths Summer Vacation Assignment Package Solution TRIGONOMETRIC RATIOS & IDENTITIES. Fundamental Relations between the Trigonometrical ratios of an angle sin + cos = or sin = cos or cos = sin + tan = sec or sec tan = + cot = cosec or cosec cot = sin cos tan = and cot cos sin sin. cosec = tan. cot = cos. sec = sin cot, cos, cot cosec cot, sec cot cot cot. Trigonometric Ratios of Compound Angles An angle made up of the algebraic sum of two or more angles is called compound angle. Some formulae and results regarding compound angles: sin (A + B) = sin A cosb + cosa sinb sin(a B) = sina cosb cos A sinb cos (A + B) = cosa cos B sina sin B cos(a B) = cosa cosb + sin A sin B. tan A tan B tan A tan(a + B) = tan A tan B, tan (45 + A) = tan A tan(a B) = tan A tan B tan A tan B, tan (45 A) = tan A tan A cot (A + B) = cot A cot B cot A cot A cot B cot B, cot (A B) = cot Bcot A sin(a + B) sin(a B) = sin A sin B = cos B cos A cos(a + B) cos(a B) = cos A sin B = cos B sin A. tan A tan B tan C tan A tan B tan C tan (A + B + C) = tan A tan B tan B tan C tan C tan A 3. Trigonometric Ratios of Multiples of an angle tan A sina = sina cosa = tan A tan cosa = cos A sin A = sin A = cos A = tan + cosa = cos A, cosa = sin A A, A

2 tan A tana tan A sin3a = 3sinA 4sin 3 A = 4sin(6 A) sina sin(6 + A) cos3a = 4 cos 3 A 3cosA = 4cos(6 A) cosa(cos6 +A) 3 3tan A tan A tan3a = tan(6 A) tana tan(6 +A) 3tan A 4. Product of sines/cosines in term of sums sina cosb = sin (A + B) + sin (A B) cos A sin B = sin (A + B) sin (A B) cos A cos B = cos (A + B) + cos (A B) sin A sin B = cos (A B) cos (A + B) 5. Sum of sines/cosines in term of products sinc + sind = sin cosc + cosd = cos C D C D cos C D C D cos sin (A B) sin (A B) tana + tanb =, tana tanb = cos A cos B cos A cos B sinc sind = cos cosc cosd = sin cot A tan A cot A cot A tan A cosec A 6. Maximum and minimum values of acos + bsin a b a cos bsin a b Hence the maximum value = a b and minimum value is a b. 7. Trigonometric Ratio of Submultiple of an Angle A A sin cos sin A C D C D sin C D C D sin or A sin cos A ve, if n sin A 4 ve, otherwise A 3 n 4 A A sin cos sin A or A 5 A A ve, if n n sin cos sin A 4 4 ve, otherwise A tan A tan a cosa + bsina a b tan A Also cosa sina = sin A = cosa 4 4

3 8. Conditional Identities : If A + B + C =, then sin (B + C) = sina, cosb = cos (C + A) cos (A + B) = cosc, sinc = sin(a + B) tan (C + A) = tanb, cota = cot(b + C) A B C cos sin, C A B cos sin C A B A B C sin cos, sin cos B C A B C A tan cot, tan cot 9. Some important identities: If A, B, C are angles of a triangle (or A + B + C = ): tana + tanb + tanc = tana tanb tanc cota cotb + cotb cotc + cotc cota = A B B C C A tan tan + tan tan + tan tan cot A B cot cot C cot A B cot cot sina + sinb + sinc = 4sinA sinb sinc C cosa + cosb + cosc = 4cosA cosb cosc A B C sina + sinb + sinc = 4cos cos cos cosa + cosb + cosc = + 4 sin. Two Simple Trigonometrical Series A B sin sin sin + sin( )+sin ( ) sin{ ( n ) }= cos +cos( )+cos( )+... +cos{ ( n ) }= C (n ) n sin sin sin (n ) n cos sin sin 3

4 . (a) cos x cos( x) x x x x cos x cos(6 x) 3 3 cos cos cos cos cos( x) cos x cos(6 x)cos x cos(6 x) cos 6 (cos(6 x) cos 6 ) cos(6 x) cos(6 x) cos 6 is independent of x. (b) (sin 7 sin ) (cos 6 cos8 ) sin sin cos(9 ) sin sin sin 3.(d) (a) tan A tan B A B C 8 A B 8 C tan(8 C) tan Atan B tan A tan B tan C tan A tan B tan C tan Atan B tan C tan Atan B tan A tan B tan C tan Atan B tanc (b) ( ) 5 3 a b c ab a b c ab a b c 3 3 cosc which is not possible. ab (c) sin A: sin B :sin C :3: 7 a : b : c : 3: 7 a k, b 3 k, c 7k since c a b then a : b : c :3: 7 is not possible 3 (d) cos Acos B sin Asin B cos (A B) = cos Acos B sin Asin B cos (A + B) = A + B = Both imply that A = 6º, B = 3º.Hence such a triangle is possible 4. (c) 3 cosec sec 3 A B = and cos sin 3 ( 3 cos sin ) sin cos sin cos sin 4 4 cos3 cos sin 3sin 4cos(3 ) 4 sin(9 5 ) cos 5 4

5 5. (a) cos 68 cos5 cos(8 8 ) 6 cos cos cos8 cos8 cos8 6. (c) sin cossin sin sin( ) sin cos sin sin (sin cos cos sin ) Dividing by sin sin sin both side we get cot cot cot tan tan tan i.e. tan, tan and tan are in H.P sin( ) cos (a) sin cos sin sin sin cos cos sin Now cos sin sin47 8.(c) tan tan 45 tan tan(8 33 ) tan 33.tan tan 45 tan cos sin cos47 tan(45 ) tan 33 tan 33 tan 33 9.(a) sin 6 cos 33 cos sin5 sin(36 4 ) cos(36 3 ) cos(8 6 )sin(8 3 ) sin 4cos3 cos6sin3 cos(36 ) cos,cos(8 ) cos, sin(8 ) sin sin(8 6 )cos3 cos6sin 3 sin 6 cos3 cos 6 sin 3 sin(8 ) sin sin(6 3 ).(c) Given xsin y cos...(i) and x sin (sin ) y cos (cos ) sin cos xsin (sin ) xsin (cos ) sin cos by (i) xsin sin cos x cos...(ii) Now by (i) and (ii) we get cos sin y cos y sin...(iii) So by (ii) and (iii) we get x y cos sin 5

6 TRIGONOMETRIC EQUATIONS Useful cases of general solutions (where n Z ). (i) sin n (ii) cos (n ) / (iii) tan n. (i) sin n ( / ) and sin n ( / ) (ii) cos n and cos (n ) 3. (i) sin sin or cosec cosec n ( ) n (ii) cos cos or sec sec n (iii) tan tan or cot cot n 4. For all: trig trig n 5. The equations of type acos bsin c are solved by transforming them to a cos cos where a b cos b, a b c sin and a b cos n n which implies that solution exists if and only if a b c a b Useful hints for solving trigonometric equations:. Factorize the equation using trigonometric formulae and identities. Each factor gives a part of solution.. Never cancel a common factor containing ' ' from the two sides of an equation. For example, consider the equation tan sin. If we divide both sides by sin, we get cos, which is clearly not equivalent to the given equation as the solutions obtained by sin are lost. Thus, instead of dividing an equation by a commong factor, take this factor out as a common factor from all terms of the equation. 3. Squaring should be avoided as far as possible. If squaring is done, check for extra solutions. For example, consider the equation sin cos On squaring, we get n sin or sin, n,,,... 3 Clearly, and do not satisfy the given equation, So, we get extra solutions. Thus, if squaring is must, verify each of the solutions. 6

7 . (b) 4 sin x sin x a (sin x) (sin x) ( a ) 4 4( a ) sin x 3 a.now 3 a 3 a 3 a 4 a a a 3 4 a (a) sin cos sin, cos n & n n & n n & n ,,,,... &, ,, n 6 4.(a) x 4x ( x ) 6 i.e. minimum value of x 4x is 6 and max. value of 3sin i.e. graph of 3sin x 4cos x is x 4cos x not intersect with x 4x i.e. no solution 5. (a) sin cos cos cos cos ( cos ) 3 3 sin cos (sin cos ) sin n 9. (b) cos x sin xy cos x sin xy cos x & sin xy x n & 3 3 xy n x n, y n. (b) 8sin x cos x cos x cos 4x sin 6 x (sin x ) sin8x sin6x cos7xsin x cos 7x 7 x (n ) x (n ) 4 7

8 QUADRATIC EQUATION. The equation ax bx c where a, and a, b, c C (Complex numbers) is called a quadratic equation. Its roots are x b b 4ac a. If and are the roots of a quadratic equation, then (a) Sum of the roots, b / a (b) Product of roots, c / a (c) Difference of the roots, D / a (d) If roots are in the ratio m:n then ( ) ( m n) mn (d) Equation formed by such given roots is x x or, x x - (Sum of roots) x + (Product of roots) = x 3. Nature of roots of ax bx c based on its Discriminant D = b If a, b, c R (Real numbers) (i) D > roots are real and distinct. D roots are real (may be equal or unequal) D = roots are real and identical. 4ac : D < roots are non real complex (imaginary) conjugates eg. + 3i, - 3i. Further If a, b, c Q (Rational numbers) (i) D is a perfect square roots are rational. (ii) D is not a perfect square roots are irrational conjugates eg. + 3, Nature of roots of ax bx c based on the sign of a, b, and c: when the sign of b matches with those of a and c, both roots are negative. when the sign of b matches with that of c only, positive root is greater in magnitude. Nature of roots Discriminant Sum of roots Product of roots One +ve, one -ve D > c/a< Both +ve roots D >.-b/a > c/a> Both -ve roots D >.-b/a < c/a> 5. (a) Condition for one common root of two quadratic equations say a x b x c and a x b x c Let the common root be, then Solving them by cross multiplication, a b c and a b c b c b c c a c a a b a b 8

9 The common root is b c bc ca ca c a c a a b a b which is also the required condition. (b) Condition for both common roots of two quadratic equations a x b x c i, a b c a b c 6. Sign of the quadratic expression (a) If D < then (i) ax bx c for all x R ax bx c when a > (ii) ax bx c for all x R when a < (b) If D >, sign scheme of y = ax + bx + c will be as follows (, Same as a Opposite in sign as that of a is i i i are roots of ax bx c ) Same as a The above result can be understood by the different possible diagrams of y ax bx c, (shown below). The portion of the curve above the x-axis is positive and that below the x- axis is negative Y Y Y (a) O a, D X (b) O X a, D (c) O X a, D Y O (d) (e) a, D Y X O X a, D (f) Y O X a, D 7. Maximum and Minimum value of a quadratic expression: From the above figure it is clear that, (a) If a >, then y = ax bx c has a minimum value. It occurs at turning point (vertex) of the parabola for which (b) If a <, then 4 x b, y ac b D. There is no maximum value. a 4a 4a y ax bx c has a maximum value. It also occurs at turning point of the curve b D which is,. There is no minimum value. a 4a 8. Useful conditions based on location of roots : (a) Condition that both roots of f(x) = ax + bx + c = will be greater than a number d is D >, d < -b/a and a.f(d) > (b) Condition that both roots of f(x) = ax + bx + c = will be less than a number e is D >, e > -b/a, a.f(e) > (c) Condition that a number g lies between the roots of f(x) = ax bx c is D >, a.f(g) < 9

10 (d) Condition that exactly one of the roots lies in the given interval (k,k ) is D >, f(k ).f(k ) < (e) If sum of coefficients i.e. a + b + c =, then one root of ax bx c is. 9. Equation of more than two degrees :- (a) If,, be the roots of 3 ax bx cx d then b b ; ; a a (b) If,,, be the roots of 4 3 ax bx cx dx c then (c) b c d ; ; ; a a a e a n n n For a polynomial equation: ax ax a x... a n d a a Sum of roots = n- Coeff.of x - ; n Coeff of x Product of roots = n Constant term - ; n Coeff. of x n. Repeated roots :- If is n times repeated root of a polymial equation p(x) =, then p x x f x n Hence, p, p''..., p.. Sign scheme of a polynomial function or rational function. Let a < b < c < d and suppose y x a x b x c x d or, y = (x-a) (x-b) (x-c)(x-d) (i) (x-b) never disturbs the sign of y as it is always positive (except at x = b). (ii) y > x (a, b) (b, c) (d, ) (iii) y < x (-,a) (c, d). Now, we concise the above discussion in following steps : Step I : Select the factor which disturbs the sign of y and find the value at which they become zero. (here, (xa), (x-c), (x-d) disturb the sign and become zero at x = a, c, d) Step II : Show these values on a number line Step III : Check the sign of y for a specific interval and sign of subsequent intervals occurs alternately. Here for x > d, y is +ve hence SIGN SCHEME of y will be as follows : ve (Note that at b (i.e. at a point between a and c, y = which is neither +ve nor -ve and be careful about it while writing the final answer) d d vc

11 . (a) Let b,4a + b + c = f ().Also a sum of roots f ( x) ax bx c for we get i.e. both the roots are real.. (b) Let f ( x) ax bx 6 then f (). Since f ( x) has no distinct real roots then f () 4a b 6 a b 3 then the least value of a + b is (b) b b 4ac b x. Now a a, b 4ac b b b 4ac and b 4ac may be or then roots are with negative real parts. b c 4. (a) a b, c are of same sign then,, & b 4ac then both the roots a a are positive, if both C and C are satisfied. 5. (c) b ac n n n n n n n n n 4 ( ) 4 ( ) ( n n) 4 3( n ) is perfect square for,, n. ( ) ( ) ( ) then 6. (b) Since a, b, c are distinct then a b c ab bc ca a b b c c a 7. (d) a 3 + b 3 + c 3 = 3abc a b c.also since roots of equation cx ax b are equal then root of the quadratic equation is ( )( 4) ( )...(ii) By (i) & (ii) 5.Now D Now for then D for.so,, 3 4, b 4ac 9 3.(c) x y y x x y x y.now 4 4y y y Now 9 y y y 4 i.e. 9 y, 4

12 COMPLEX NUMBER. Definition of a complex number : A number of the form z x iy, where x, y R and i is called a complex number. x Re( z), is called real part of z and y Im( z), is called imaginary part of z. z x iy is called (a) real if y (b) imaginary if y (c) purely imaginary if x On this basis is a real and purely imaginary number as well.. Equality of two complex number : (a) x iy a ib x a and y b i.e. real and imaginary parts are separately equal. (b) Inequality does not hold in a complex plane. i.e. 3 4i 3i has no sense. 3. Representation of complex number : (a) Algebraic form : z x iy. (b) Ordered pair form : z = (x,y). z is also represented on a plane as a point (x,y). Its real part is shown on the x-axis (real axis) and imaginary part on the y-axis (imaginary axis). This plane is known as Complex plane or Argand plane or Gaussian plane. (C) Polar form or trigonometric form : z = r (cos + isin ) where r = z and = arg z. (e) Euler or Exponential : z i re 4. Three basic terms of a complex number (a) Modulus: If z x iy, its modulus, is denoted by Y O r z x y P x, y Real axis X Geometrically, it is the distance of point z from the origin. (b) Argument (or amplitude) : Geometrically, it is angle made by the line joining the origin to the point z(x, y), with x-axis. If z = x + iy, then arg( z) tan / y x ( z being in first quadrant) arg( z) tan / y x ( z being in second quadrant) arg( z) tan / y x ( z being in third quadrant) arg( z) tan / y x ( z being in fourth quadrant) (c) Conjugate of a complex number : If z = x + iy, then its conjugate is denoted by z x iy. Geometrically, it is the reflection of point z with respect to x-axis.

13 5. Properties of modulus, argument and conjugate :- n z z z z ; z z i. ii. Triangular Inequalities: n ; z z z z z z z z z z and z z z z z z z z z z and z z z z hold if and only if Note: Equalities collinear and are on the same side of the origin. z and z are iii. z z, z z ; z z z z ; zz zz ; z z z z ; zz z z z z z iv. Re( z) ; Im( z) ; z Re z z ; z Im z z i v. If z z, then the complex number is real. If z z, then it is imaginary. or z z z z z z cos arg z arg z z z z z Re( z z ) arg z z arg z arg z vi. n ; arg z n arg z ; arg z / z arg z arg z vii. arg z arg z ; arg kz arg z if k and 6. Cube roots of unity : arg kz arg z if k 3i 3i (a) Solving 3 x, we get: x,, ; called cube root of unity, denoted by, &. (b) Properties of cube roots of unity i. Their sum i.e. 3 and their product i.e. ii. If we square one of the complex cube root of unity, we get the other. iii. Their modulus i.e. ; their arguments are, / 3 and / 3. n n iv. 3 or according as n is a multiple of 3 or not. v Cube roots of unity are vertices of an equilateral traingle which is inscribed in a circle of unit radius with centre at origin. 7. De Moivre s Theorem : To find rational powers of a complex number (Polar form is must) n (a) If n is an integer then (cos isin ) cos n i sin n (b) If n is an integer then 8. Euler s theorem : i cos isin e, Hence, we have / (cos sin ) cos k i n i sin k where k=,,,...,n-. n n cos A B C i sin A B C (a) (cos A i sin A)(cos B i sin B) /(cosc i sin C) (b) cos isin cos isin 3

14 9. n th roots of unity : i. If we consider e i n, then n th 3 n roots are,,,,...,. So n th roots of unity are in G.P. ii. Sum of n roots of unity =, and Product of n roots of unity = ( ) n Sum of n th roots of a complex number z = ; Product of the n th roots of z = ( -) n+ z iii. Modulus of each of n roots = ; their argument, / n, 4 / n,...,[( n ) / n] are in A.P. iv. These n roots of unity lie on a unit circle as vertices of a regular polygon of n sides.. Square roots of a complex number x iy x iy x y x,xy y. Solve for x and y. (i) / (,) ( / n) e (,) n r cos i sin (ii) / o o o. Geometrical relations ro o o cos isin (a) If z and z are two complex numbers, then the complex number and z in the ratio m : n.. Here, use tan( / ) ( cos ) / sin ; nz mz z m n (b) z-z z-z is equation of perpendicular bisector of line joining the points z and z. divides the join of z (c) Different forms of Equation of a circle are z z r, represents a circle with radius r and centre i. ii. zz az az b, (where b is a real number), represents a circle with centre -a and radius a b iii. z z z z arg ( ) iv. z. represents arc of a circle through z and z. z-z k z-z ( if k ) is also a circle. Ends of diameter (k : ) and center (- k :). (d) The triangle whose vertices are represented by z, z, z 3 is equilateral if and only if or z z z z z z z z z z z z z z z 3 3. Concept of rotation : (a) Complex number i ze is obtained by rotating Oz with an angle in anticlockwise direction. It follows directly from the geometrical meaning of multiplication of two complex number. (b) In general, we have z3 z z 3 z z z z z cos i sin z 3 z z O 4

15 5 5.(a) i 5 can be written as i i i i Hence real part of 5 i i i i.. (b) 4(5 5 i) i z z (5 5 i) z. Now () i 5 4 i 3.(c) z i i i 5 5 (5 ) then (5 i) 6 i z or or i.then the product of the real part of the roots of z z = 5 5i is ( 6). z x iy x iy (if z x iy ) x y xyi x iy x y x xy y i.by equality of complex no. x y x... () and xy y y x y or Put y in () we get x x x x x. ( ) x, 4 i 3 i and put x in () we get y i.e. 4 y which is not posible since y is real. 4 hence, z and i i.e. number of solutions is. 4.(c) Given equation x p iq x i ( ) 3.If & are the roots then p iq & 3i. Given 8 8 ( p iq) 6i 8 p q i( pq 6) 8 comparing real & imaginary parts, or p 3& q p q 8 & pq 3.On solving we get p 3& q 5.(a) It is given that for a complex no. z x iy 6.(c) then Re z Im z represents x y which is the equation of square i 5 3 4i. 3 4i 3 4i 3 4i 3 4i 9 6 Now by figure we see that opposite angles are supplementary. Hence it is a cyclic quadrilateral. 7.(d) For a complex no. z it is given that 3 z z z ( z z ) z or Hence z ; z, z i.hence no. of roots are 5. - x + y = - x - y = ( 3,4) x + y = x - y = 5 (,) z (i).takiing modulus of both sideswe get 3 z (,) z 5 z z or zz z or 4 z by (i) (3,4) 5

16 8.(c) Let F is x iy then reciprocal of F is 9.(a). (c) x iy x iy x y i.e. reciprocal number is canjugate and its modulus is less then since modulus of F is greater than i.e. C is required number. we have, z 5i z 5i z 5 i z ( 5 i) z is equidistant from 5i & -5i z lies on the perpendicular bisector of line segment joining (,5) & (, 5) i.e. x -axis z z z z z z z z z z cos( ) z z z z cos( ) i.e. Arg z Arg z = 6

17 STRAIGHT LINE. Coordinate Systems: Cartesian and Polar Systems (a) x = r cos, y = rsin (b) r x y, = tan - (y/x) (c) x coordinate is also called abscissa and y coordinate is called ordinate. Y P x, y or r, X. Distance between two points (x, y ) and (x, y ) is x x y y 3. Section Formula: If a point P divides the line segment joining points (x, y ) and (x, y ) in the ratio (a) m : n internally, then mx nx my ny P, m n m n (b) m : n externally, then mx nx my ny P, m n m n P m P( x, y) (c) If P is the mid point, then x x, y P y n P (d) The ratio in which straight line by c ax divides the line segment joining the points x ax by c x, y is = ax by c. Thus, points (x, y ) and (x, y ) are on the same side (or opposite sides) of the line ax by c according as ax by c and ax by c have same sign (or opposite sign). 4. (a) Area of a triangle whose vertices are (x, y ), (x, y ) and (x 3, y 3 ) is, y and x y x y x y 3 3 x y x3 y 3 x y = x x x y y y 3 3 (b) Area of a polygon whose consecutive vertices are (x i, y i ) (i =,, 3,...n) is x y x y x y x y n n n n... x y x y x y x y 3 3 n n 7

18 5. (a) Centroid : The point (G) where medians of the triangle meet. We must note that (i) AG BG CG GD GE GF F A x, y G E (ii) x x x3 y y y3 G, 3 3 (b) Incentre : is the point (I) where internal bisectors of the angles meet. We must note that (i) (ii) BD AB c etc. DC AC b ax bx cx3 ay by cy3 I, a b c a b c (c) Excentres : are points where angle bisectors of one internal and two external angles meet. ax bx cx3 ay by cy3 Excentre opposite to A, a b c a b c etc. (d) Circumcentre : The Point (O) where perpendicular bisector of the sides of the triangle meet. It is centre of the circle passing through the vertices. (i) AO = BO = CO = R, called circumradius BOD COD A (ii) (iii) Slope of OD x Slope of BC = - etc. (e) Orthocentre : is the point (P) where altitudes of the triangle meet. We must note that : Slope of AD x Slope of BC = - etc. 6. Properties of centres : (a) In an equilaterial triangle, the four centres (centroid, incentre, circumcentre and orthocentre) are coincident. (b) In a right angled triangle ABC A 9 (c) (d) orthocentre is at A and circumcentre is the mid point of hypotenuse BC. Lines joining the centroid and vertices of a triangle ABC, divide the triangle into three equal areas. Orthocentre (P) centroid (G) and circumcentre (O) of a triangle are collinear and A B D C x, y x 3, y 3 A x y F, P x y x3, y3, B D E C AP PG AG P G also, OD GO GD O B C D 7. Slope of a line segment : If a line makes an angle with x-axis in anticlockwise direction, then tan is called slope of the line. It is generally denoted by m. Instead of x-axis, if the line makes an angle with positive direction of y-axis, then slope = tan Slope of a line joining the points (x, y ) and (x, y ) is = y y x x. 8

19 8. Collinearity of points A, B, and C: (a) Slope of AB = Slope of BC (= Slope of CA) (b) Area of triangle ABC = (c) A divides BC in some ratio, i.e. Section formula holds. (d) Sum of two of AB, BC, and AC is equal to the third. 9. Locus : If a point moves such that it follows some (geometrical) condition, then the path traced out by the point is called its locus, and mathematical relation thus obtained is called equation of the locus. It is generally an equation connecting the coordinates (x and y) of the point and the given constants.. Trasformation of Coordinate System: General equations of transformation of coordinate system: On shifting of origin to point (h,k) and rotation of axes by an angle, the old coordinates (x,y) and new coordinates (x,y ) of a point are related by: x x 'cos y 'sin h y x 'sin y 'cos k ; which become x x ' h y y ' k. Various useful forms of a straight line: Equation of a straight line which is in case of only translation of axes (i.e. no rotation) (a) Parallel to x-axis is y = c (a constant) Parallel to y-axis is x = c (a constant) Eq. of x-axis : y = ; and that of y-axis : x =. (b) Passing through origin, having slope m, is y = mx (c) Having slope m and intercept on y-axis c is y = mx+c (d) Having slope m and passing through (x, y ) is y - mx = y - mx (e) Passing through points (x, y ) and (x, y ) is y y x x (f) Having intercepts a and b on the axes respectively is x y a b (g) If length of perpendicular from the origin to a line is p and the perpendicular line (OP) makes an angle with the positive y x y x direction of x-axis, then the equation of the straight line (AB) is x cos y sin p A B C (h) (i) Distance form or paramatric form of a straight line is A point at a distance r from the point (x, y ) on the straight line can be taken as x, y r cos x, r sin y Equations of straight lines (PA and PB in the figure) passing through the point (x, y ) and makin an angle with the straight line of slope m tan x x y y r cos sin are y y tan x x For (g) For (h) For (i) 9

20 . General equation of a straight line is ax+by+c=. Its slope = -a/b, and its distance from origin = m m 3. Angle between two straight lines: whose slopes are m and m is, tan m m Acute and obtuse angle between them are tan m m m m and tan m m m m respectively. If one line is parallel to y-axis and slope of the other line is m then angle between them is tan 4. Parallelism and Perpendicularity: Two straight line y = m x + c and y = m x + c are (a) parallel if m = m slopes are equal (b) perpendicular if m m = - i.e. product of slopes = If a i x + b i y + c i = (i =, ) are two lines and (a) (c) a a b a b c b lines are intersecting. (b) a b c lines are parallel. a b c a b c lines are coincident. (d) a a b b lines are perpendicular. 6. (a) Equation of a straight line parallel to the line ax by c can be taken as ax by k. (b) Equation of a straight line perpendicular to the line ax by c can be taken as bx ay k. a c b m 7. Perpendicular distance of point (x, y ) from the line ax by c is ax by c a b d c Distance between two parallel lines ax by c and ax by d is a b 8. Concurrency: Three lines L a x b y c (i =,, 3) are concurrent if (a) (b) i i i i Point of intersection of any two lines satisfies the third one, or a b c a b c a b c Equation of a family of lines (or a variable line) passing through the point of intersection of the lines L = a x + b y + c = and L = a x + b y + c = is L L. Equation of bisectors of the angle between the straight lines a x b y c and x b y c a are a x b y c a x b y c a b a b If the signs of c andc are kept same then for origin containing angle bisector, " " sign is taken; and for acute angle bisector, sign opposite to that of " aa bb '' is taken. x xo y y axo byo c o. Foot of perpendicular from (x,y ) to the line ax+by+c = is given by: a b a b.

21 . Question incomplete. (c) Slope of AB and 4 AB 4 5 then BC 5. Since D is centre of square then slope of CD and CD 5 4 coordinate of C is, (,) (,) B 45 5 (,)C 5 D O A(4,) Now equation of CD is x y 5 coordinate of D is x 3, y (a) Slope of AB x then tan ABX x A(,) C (5,3) slope of 3 BC 5 x then tan CBX 3 5 x X' B ( x,) X 3 Now x 3 3x 5x 3.Now slope of x 5 x 5 AB So equation of AB is 5x 4y 3. 5.(c) Equation of the median through A is (px + qy -) (qx + py - )=.Since it passes through (p, q) then p q pq then the equation of the median is(pq )(px + qy ) = (p + q ) (qx + py). 6.(b) 7 Coordinates of point A is (3,).Coordinate of point B is (,4).Coordinate of point P is, 5 5 Now slope of PA is 7 3,slope of 96 4 PB is 8 4.Since then PA PB then AB is diameter of PAB i.e. circumradius 5

22 7.(a) Let AB x p y q r then equation of line AB is r x p r cos, y q r sin will lie in cos sin ax by c then ap ar cos bq br sin c r ap bq c acos bsin AB ap bq c a cos bsin 8. (a) If origin is shifted to ( h, k ) then new coordinates of ( x, y ) is ( x h, y k). So the triangle formed by the points (x + h, y + k), (x + h, y + k) and (x 3 + h, y 3 + k) is congruent to the triangle formed by the points (x, y ), (x, y ) and (x 3, y 3 ) then A A'..(a) Equation of bisectors of angle formed by k u k v = and k u + k v = for non zero real k and k are ku kv ku kv k k k k or ku kv k k ku or kv u, v then equation of the bisectors of angle formed by is uv

23 CIRCLE. Circle is locus of a moving point such that its distance from a fixed point remains fixed. General equation of second degree in x and y, ax hxy by gx fy c represents a circle if a = b and h =.. Equation of circle in various forms : (i) Central form: x h y k r has centre (h, k) and radius r. eg. circle with centre circle with centre h, k and touching x-axis is x h y k k h, k and touching y-axis is x h y k h (ii) General equation: x y gx fy c has centre (-g, -f) and radius g f c the circle will be a real circle, point circle or imaginary circle depending on whether g f c, or ; the circle passes through the origin if c ; Length of the intercept made by the circle x y gx fy c on x-axis is g c and that on y-axis f c. (iii) Diametric form: Equation of the circle having x, y and, y (iv) x x x x y y y y Parametric form: x as extremities of a diameter is A(x ), y For circle x y r : x r cos, y r sin x h y k r : x h r cos, y k r sin For Parameter is such that. B(x y ), (x,y) 3

24 3. Position of a point w.r.t. a Circle: (i) Condition that point x, y lies inside, outside or on the circle is S x y gx fy c,, respectively. S x y gx fy c (ii) The greatest and least distance of a point P from a circle with centre C and radius r is PC r and PC r (iii) For a point on the circle: Equation of the tangent at any point (x,y ) on the circle S= is T = T xx yy g x x f y y c ). This is called as Point form of tangent. (where (iv) For a point inside the circle: Eq. of chord of a circle S =, whose middle point is (x, y ) is T = S. (v) For a point outside the circle: (a) Length of tangent from a point outside the circle to the circle S = is S. Here, the square of the length of tangent, i.e. S, is called the Power of this point. (b) Angle between these tangents tan r / S (c) (d) If two tangents PT and PT are drawn from point P to a circle S = then the equation of pair of tangents is SS = T. The equation of chord of contact (T T ) of tangents from the outside point (x, y ) to the circle S= is lr T=; and the length of the chord of contact is where r is the radius of circle and l is the l r length of tangent. r S (x,y ) 4. Line and a Circle: (i) If the distance of centre of circle is p then p p p r r r with radius r from the line lx my n x y gx fy c the line and circle have no common point the line touches the circle. This is condition of tangency. the line intersects the circle at distinct points, i.e. it is a secant. p the line is a diameter. the intercept made by circle on this line = r p and the angle subtended by the chord on the centre of the circle = cos p / r (ii) Line y mx c touches the circle x y a if c a m x h y k a is Thus, the tangent having slope m to the circle y k m x h a m h am a, k m m ( ) whose point of contact is This is called as Slope form of tangent. (iii) Normal at the point (x, y ) on the circle S = is a line passing through its centre. r p 4

25 6. System of two Circles (with centres, C and C,distance d apart, and radius r and r ) (i) do not touch each other (nor lie inside) if d > r + r ; they have 4 common tangents. Direct common tangents are intersecting at P and transverse common tangents are intersecting at P, as shown in the figure. PC r P C PC r P C C r d r P C P (ii) touch each other externally if d = r + r ; and have 3 common tangents. (iii) intersect at two distinct point if r - r < d < r + r and number common tangent is. For the angle of intersection: r r d cos r r Two circles x y gi x fi y ci, (i =,) cut orthogonally i.e. 9 if g g + f f = c + c (iv) touch each other internally if d = r - r ; and have only common tangent. (v) one lies inside other if d < r - r. 7. Radical Axis: It is the locus of a point which moves so that the length of tangents drawn from it to two circles S =, and S =, are equal, is given by S-S =. This is same equation as that of the common chord if the circles are intersecting at two points, or that of the common tangent if the circles are touching. 8. Family of the circles: (i) (ii) Equation of any circle passing through points of intersection of the circle S = and S = is S S, where Equation of any circle passing through points of intersection of the circle S = and the line L = is S L 9. Other Useful Concepts: (i) Director circle: The locus of a point from which perpendicular tangents are drawn to the circle x +y = r is x +y = r. (ii) Pole and Polar: The equation of Polar of pole P(x,y ) w.r.t. a circle S = is T =. Polar of Pole P w.r.t. a Circle is the locus of points of intersection of tangents which are drawn at the points where a variable line passing through the Pole meets the Circle. (iii) Two lines a x b y c, and ax b y c cut the coordinate axes in concyclic points if a a b b. 5

26 . (b) Let the circle be ( x h) ( y k) r...(i) which touches the circles externally x y a...(ii), ( x a) y ( a)...(iii) then h k r a...(iv) ( h a) k r a...(v) h a k h k a By (v) - (iv) 4( h k ) 9a 4ah 6h Hence the locus of centre is x 4y 4ax 9a or ( x a) 4y 3a. (a) Equation of a circle passes through (,),(,) and (, ) is x y x y. 5 It also passes through (k, 3k) then 3k 5k k, i.e. for two values of k (c) r r r AB BC r 3 As figure, 3 r r 3 then the radii of c and c are &. 4. (b) Y As figure h b k a h k a b b b h h + b ( h, k) then the equation of the locus of the centre of the circle passing through the extremities of the two rods is x² y² = a² b². O k + a a k a X 8. (c) Let equation of circle S x y gx fy c. Now common chord of circle S & x y 4 is gx fy c 4 which will pass through (,) since S bisect the circumferences of the circle x y 4 then c 4 c 4. Also common chord of circle S & x y x 6y is (g ) x ( f 6) y c which will pass through (, 3) since S bisect the circumferences of the circle then (g ) 3( f 6) 4 g 6 f 5. The locus of the centre of the circle which bisects the circumferences of the circles is x 6y 5 i.e. a straight line. 6

27 9. (b) BOD BAO (5 ) 3.Now B 3a a DO a cos3, BD asin 3 then coordinates of B & C are a D 3 3 a o 5 A 3 a a 3 a a, and, C. (a) Distance between centres is 4 r r. Hence they touch and common tangent by S S is x i.e. y axis. Other two direct common tangents intersect at x = C (, 3) 3() ( 3) 3() () x, y x, y 3, 3 3 C (-3,) (,) A (3,) and its equations are found as x 3y 3 and x 3y 3. These two tangents meet the B (,- 3) common tanget x at Q(, 3) and R (, 3).Clearly ABC is equilateral. Hence centroid of PQR is (, ). 7

28 PARABOLA Locus of a point P which moves such that its distance from a fixed point S (called focus) bears a constant ratio e (called eccentricity) to its distance from a fixed line (called directrix), and where e PS = PD, where D is the foot of perpendicular from P to the fixed line.. Standard equation of a Parabola: y 4ax (i) Focus, S = (a, ), Vertex is (, ), Equation of axis y, Equation of tangent at vertex x, Equation of directrix x a. (ii) Focal chord of a parabola : Any chord passing through focus of the parabola. Latus rectum : A chord passing through focus perpendicular to the axis of parabola. Its equation is x = a. Length of latus rectum = 4a = 4x distance between focus and vertex. End points of latus rectum are La,a and L' a, a (iii) Double ordinate : A chord which is perpendicular to the axis of parabola. (iv) Focal distance of any point P( x, y ) : PS x a K. Point x, y will be outside, on or inside the parabola depending on whether y negative. 3. Parametric equation of y 4ax Equation of tangent at a point (at, at) and Equation of normal: x at y ty x at at y xt at at 4. Equation of tangent at a point (x, y ): yy a x x Equation of tangent in terms of slope m: 3 4ax is positive, zero or i.e. T a y mx m where the point of contact (a/m, a/m) 8

29 5. Equation of normal at P x, y y y x x and equation of normal in terms of slope m: y a y mx am am where the foot of normal is (am, - am) 6. Properties of normals Number of normal : y = mx - am - am 3 is a cubic equation in m. So in general at most three normals can pass through a point. The sum of slopes of these normals is zero and also sum of the ordinates of the feet of normal is zero. 7. Properties of focal chords: Tangents at the extremities of any focal chord intersect at right angles on the directrix tt 8. Other Parabolas: To solve problems related to parabolas other than y 4 like shifting of origin etc, e.g. y 4a x is, and whose axis is parallel to x-axis. x 4a y 3 ax, we have to use concepts is the equation of that parabola whose vertex is the equation of that parabola whose vertex is, and whose axis is parallel to y-axis. 9. Other important points: Point of intersection of tangents at t and t is at t, a( t t ) i. e. (GM, AM) Point of intersection of normal at t and t is { a a( t t t t ), at t ( t t )} This point will be the parabola if t t If the normal at the point (at, at ) meets the parabola again in the point (at, at ), then t t ( / t ) Equation of pair of tangents of a parabola drawn from a given point (x, y ) is T SS, Chord of contact T, Polar T Chord of parabola whose mid point is (x, y ) is T S PD SP ST SN, DPT SPT, PSR PKS 9 9

30 . (d) Length of semilatus rectum is H.m. of two segments of a focal chord then. (c) Coordiinate of P & Q are, at at and ( at, at ) (3)(5) 5 a a P Now slope of OP is t and slope of OQ is t then. t t O R t t...(i) 4 Now equation of line PQ is x ( t t ) y at ( t t )at att Q Now by (i) x ( t t) y 8a. So coordinate of point R is (4 a,) i.e. OR 4a. 3. (c) Any tangent to Any tangent to y 4( x ), a is y m( x )...(i) m y 8( x ); a is y M ( x ) M but tangents to the parabolas are perpendicular to each other then Mm M / m i.e. y ( x ) m...(ii) m On subtracting we get m x 3 m m m 4. (a) Equationof circle taking OP as diameter is x x at y y at x y at x at y or x 3 ( at, at ) Similarly equation of circle taking OQ as diameter is x y at x at y a t t x a t t y The equation of OR is ( at, at ) R t t ( t t) x y tan t t tan...(i) Now equation of tangent at point ( at, at ) is yt x at i.e. t tan cot t. Similarly cot t.by (i) we get cot cot tan 5.(a) If one end of focal chord of parabola y 4x is t,t then coordinates of ther end is, t t then length of focual chord is 4 t t t t 4 t t t t t t t t 3

31 6. (d) One end of normal chord of parabola y 4ax is P( at, at ) then other end is Q a t,a t t t.now slope of OP then slope of OQ is t t t Since the normal chord at a point ' t ' subtends a right angle at the vertex therefore. 4 t t t t t 8. (d) Slope of t SP t,slope of t SQ t since PSQ. Now 45 P( at, at) a t tan 45 t t at S (a,) M( at, ) t t t tangent at P is yt x at, tangent at Q is yt x at Now intersection pont of tangents is at a,, 45 Q ( at, - at) 9. (c) Normal at point t,4 t cut parabola again at t,4 t t t then 4t 4 t t t t then ends of normal are 8, 8 and 8, then slope of the normal chord is.. (a) In a MDS, at a MDS at 3, tan 3 t 3 M (- a, at) 3 P( at, at) then coordinate of P is 3 a,a 3 D S(a,) Now SP 3a a a 3 4a 3