HYPERBOLA HYPERBOLA A Conic is said to be a hyperbola if its eccentricity is greater than one. If S ax22 hxy by 2 2 gx 2 fy c 0 represents a hyperbola then h2 ab 0 and 0 2 2 If S ax2 hxy by 2 gx 2 fy c 0 represents a rectangular hyperbola then h2 ab 0 , 0 and a b 0 x2 y 2 Hyperbola: Equation of the hyperbola in standard form is 1. a2 b2 Hyperbola is symmetric about both the axes. Hyperbola does not passing through the origin and does not meet the Y - axis. Hyperbola curve does not exist between the lines x a and x a . x2 y 2 xx yy x2 y 2 x x y y Notation: S 1 , S1 1 1, S1 1 1, S1 2 1 2 1. a2 b 2 1 a2 b 2 11 a2 b 2 12 a2 b 2 Rectangular hyperbola (or) Equilateral hyperbola : In a hyperbola if the length of the transverse axis (2a) is equal to the length of the conjugate axis (2b) , then the hyperbola is called a rectangular hyperbola or an equilateral hyperbola. The eccentricity of a rectangular hyperbola is 2 . Conjugate hyperbola : The hyperbola whose transverse and conjugate axes are respectively the conjugate and transverse axes of a given hyperbola is called the conjugate hyperbola of the given hyperbola. 2 2 1 x y The equation of the hyperbola conjugate to S 0 is S 1 0 . a2 b 2 If e1 and e2 be the eccentricities of the hyperbola and its conjugate hyperbola then 1 1 2 2 1. e1 e2 x2 y 2 Parametric Equations: The parametric equations of a hyperbola 1 are a2 b 2 x asec and y b tan where is parameter ( 0 2 ).A point on the hyperbola is represented by ( asec , b tan ) ¨ Auxiliary Circle : The circle described on the transverse axis of a hyperbola as diam- eter is called the auxiliary circle of the hyperbola The locus of the feet of the perpendiculars from foci on any tangent of the hyperbola is a circle concentric with the hyperbola. This circle is called Auxiliary circle of the hyperbola. The equation of auxiliary circle of S=0 is x2+y2=a2 Director Circle: The locus of point of intersection of perpendicular tangents to a hyperbola is a circle concentric with hyperbola. This circle is called director circle of the hyperbola The equation to the director circle of S 0 is x2 y 2 a 2 b 2 1 SKVVMATH.COM HYPERBOLA x2 y 2 x2 y 2 1 1 Characteristics a2 b2 a2 b 2 a2 b 2 a2 b 2 Ecentricity e e a2 b2 Center (0 , 0) (0 , 0) Vertices ( a,0) (0,b ) Foci ( ae,0) (0,be ) 2 2 b 2 2 a Length of latusrectum a b b 2 a 2 Ends of latusrectum ( ae, ) ( , be ) a b y be Equation of latusrectum x ae Length of transversal axis 2a 2b Length of conjugate axis 2b 2a a b Equation of diretrices x x e e Equation of transversal axis y 0 x 0 y 0 Equation of conjugate axis x 0 2 2 2 2 2 2 Equation of auxillary circle x y a x y b 2 2 2 2 2 2 2 2 Equation of director circle x y a b x y b- a The absolute difference of focal distances of any point on the hyperbola 2a 2b 2 SKVVMATH.COM HYPERBOLA Position of a Point with respect to a hyperbola: Let P(x1 ,y 1 ) be a point and x2 y 2 S 1 0 be a hyperbola then a2 b 2 (i) p lies on the hyperbola iff S11 0 . (ii) p lies inside (i.e. in the region not containing the centre) the hyperbola iff S11 0 . (iii) p lies out side the hyperbola iff S11 0 . Tangent to Hyperbola: The equation of the tangent at p(x1, y1) is S1 0 . x y The equation of the tangent at " " is sec tan 1. a b The condition for a straight line y mx c to be a tangent to the hyperbola S 0 is a2m b2 2 2 2 2 c a m b and the point of contact is , . c c The equation of the tangent to the hyperbola S 0 having slope m is b b 2 2 2 ,m, m y mx a m b a a Two tangents can be drawn to a hyperbola from an external point. x2 y 2 If m,m are the slopes of the tangents through P(x y ) to the hyperbola 1 1 2 1, 1 a2 b2 2x y y 2 b2 m m 1 1 m m 1 then 1 2 2 2 , 1 2 2 2 . x1 a x1 a The product of the perpendiculars from foci to any tangent to the hyperbola S 0 is b 2 . Normal to Hyperbola: The equation of the normal to the hyperbola S 0 at (x1 , y 1 ) is 2 2 a x b y 2 2 a b . x1 y1 ax by 2 2 The equation of the normal at is a b .0, , sec tan 2 The condition that the line lx my n 0 may be normal to the hyperbola S 0 is a 2 b2 (a2 b2 )2 . l 2 m2 n2 The condition that the line y mx c to be a normal to the hyperbola S 0 is m a2 b 2 c a2 b 2 3 SKVVMATH.COM HYPERBOLA x2 y 2 m a2 b 2 Equation of normal to the hyperbola 1 in slope form is y mx a2 b 2 a2 b 2 Co-normal points: In general four normals can be drawn to a hyperbola from any point. If ,,, be the eccentric angles of these four co-normal points then is an odd multiple of i.e., Sum of the eccentric angles of four co-normal points is an odd multiple of . If ,, are the eccentric angles of three co-normal points on the hyperbola x2 y 2 1 , the normals at which are concurrent, then a2 b 2 sin sin sin 0 x2 y 2 The equation of the chord joining the two points and on the hyperbola 1 a2 b2 is x y cos sin cos a2 b 2 2 x2 y 2 If P x, y a sec , b tan is a point on 1 and SS, are foci then. 1 1 a2 b 2 SP ex1 a aesec a and S P ex1 a aesec a x2 y 2 If Px , y is a point on 1 0 and SS, are foci then SP ey b; S' P ey b 1 1 a2 b 2 1 1 Asymptotes : The tangents at infinity to the hyperbola are called asymptotes of the hyperbola. x2 y 2 The pair of equation of the asymptotes of the hyperbola S 0 is 0 . a 2 b2 The equation of a hyperbola and its pair of equation of asymptotes always differ by a constant. The asymptotes are always passing through the centre of the hyperbola. Asymptotes are equally inclined to the axes of hyperbola. 1 b The angle between the asymptotes of the hyperbola S 0 is 2sec1(e) or 2 tan . a The product of the perpendiculars from anypoint on the hyperbola to its asymptotes is a2b2 . a 2 b2 Any straight line parallel to an asymptotes of a hyperbola intersects the hyperbola in only one point. Asymptotes of a hyperbola and its conjugate hyperbola are same If SS, ' and A are the equations of a hyperbola, its conjugate hyperbola and its pair of asymptotes respectively then SS' 2 A 4 SKVVMATH.COM HYPERBOLA Rectangular Hyperbola x2 - y 2 = a 2 : The hyperbola in which the lengths of the transverse and conjugate axes are equal is called an equilateral or rectangular hyperbola ( a b ) (i) Eccentricity, e 2 (ii) The equation of the asymptotes are y x (iii) The angle between the asymtotes is 90 . Rectangular hyperbola xy = c2 : If we rotate the axes through an angle of 450 in clock wise direction x2 y 2 a 2 will 2 2 a become xy c2 , c . (i) centre 0,0 (ii) vertices c, c (iii) Foci 2 2c , 2 c (iv) e 2 (v) LLR 2 2 c (vi) axes y xy, x (vii) Asymptotes x0, y 0 (viii) Angle between the asymptotes 900 2 (ix) Equation of tangent at x1, y 1 is xy1 xy 1 2 c 2 2 (x) Equation of nomral at x1, y 1 is xx1 yy 1 x 1 y 1 c (xi) Parametric equations xcty, , t R 0 t (xii) Equation of tangent at 't ' is x yt2 2 ct 0 3 4 (xiii) Equation of noraml at 't ' is xt yt ct 1 3 (xiv) Normal at t1 meets the curve again at t2 then t1 t 2 1 c c c 2 (xv) Pct1,,,,, Qct 2 Rct 3 are points on xy c then the ortho centre of PQR t1 t 2 t 3 c is S, cttt1 2 3 ttt1 2 3 Note : S also lies on xy c2 xiv) If a circle and a rectangular hyperbola xy c2 meet at four points c 4 4 xyii, ct i , , i 1,2,3,4 then tttt1 2 3 4 1 xxxx1 2 3 4 c, yyyy 1 2 3 4 c ti c c c c Note: Pct1,,,,,,, Qct 2 Rct 3 Sct 4 are concyclic then the ortho centre of t1 t 2 t 3 t 4 c PQR ct4 , S t4 The hyperbola conjugate to xy c2 is xy c2 The equation of the hyperbola having x and y as asymptotes can be taken as x y k 2 5 SKVVMATH.COM HYPERBOLA Equation of the hyperbola with reference to two perpendicular lines ax1 by 1 c 1 0 and 2 2 axbyc bxayc 1 1 1 1 1 2 2 2 2 2 bx ay c 0 as axes of coordinates is a b a b 1 1 2 1 1 1 1 1 a2 b 2 The equation to the chord of contact of Px 1, y 1 with respect to the hyperbola S 0 is S1 0 .