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Pure Maths/Hyperbola/p.1 Hyperbola focus : (ae, 0) a For the standard form of the equation of the ellipse, directrix : x = e or Let P(x, y) be a variable point on the hyperbola. Distance of (x, y) from the focus (ae, 0) a Distance of (x, y) from the directrix x = e or y P(x, y) x = e Distance of (x, y) from the focus (- ae, 0) a Distance of (x, y) from the directrix x = e (-a, 0) x x F’(-ae,0) = e x - ae a + y2 = e2 - x e 2 x2 1 - e 2 + y2 = a 2 1 - e 2 x2 y2 = 1 a 2 a 2 e2 - 1 Principal focus x=a a e x=- e x2 y2 = 1 2 2 a b L y x B(0,b) x2 y2 = 1 a2 b2 (a, 0) x x x F(ae, 0) 0 Secondary focus where e > 1. 2 focus : (- ae, 0) directrix : x = - a e x where b2 = a 2 e 2 - 1 V’(-a,0) V(a,0) x x 0 x F(ae,0) . x B’(0,-b) Consider L’ PF - PF' = a a e x - - e x - - e e = 2a ∴The locus of a variable point moving in such a way that the absolute value of the difference of its distances from 2 fixed points (foci) is a constant is a hyperbola. Remarks 1. V, V’ - vertices of the hyperbola ; VV’ - transverse axis (a) (b) x2 y2 = 1 32 22 y2 x2 = 1 32 22 ; BB’ origin 0 - centre of the hyperbola - conjugate axis ________________________________________________ (c) x2 y2 = 1 22 32 Pure Maths/Hyperbola/p.2 (d) 2. y2 22 - x2 32 = 1 The chord through the focus and perpendicular to the transverse axis - latus rectum (rectara) (ae)2 y2 = 1 a2 b2 ∴ b4 y2 = b2 e 2 - 1 = a2 2b 2 length of latus rectum = a 2a 2 e 2 - 1 LL' 2b2 1 = = AF a a(e - 1) a 2 e - 1 = 2 (1 + e) > 4 3. Point of intersection of the transverse axis and conjugate axis is called the centre of the hyperbola. 4. Asymptotes of Hyperbola : y x2 y2 = 1 a2 b2 x2 y2 = 0 a2 b2 Equation of two asymptotes : x y = 0 a b 5. and x y + = 0 a b b2 = a 2 e 2 - 1 ∴ 6. x y x y x - + = 0 a b a b b=a iff e= 2 The equation of the hyperbola becomes x2 - y2 = a 2 and such a hyperbola is called the standard form of rectangular hyperbola whose asymptotes are perpendicular to each other. y Parametric Equation : x P (a) Trigonometric Parametric Equations x2 y2 = 1 a 2 b2 x = a sec y = b tan a 0 x x where is the eccentric angle and 0 < 2 . Auxiliary circle : x2 + y2 = a2 Pure Maths/Hyperbola/p.3 (b) Algebraic Parametric Equations x = a sec y = b tan Let t = tan 2 Ellipse : 1. b2 = a 2 1 - e 2 2. a 1 + t2 x = 1 - t2 ; y = x2 y2 + = 1 a2 b2 x + my + n = 0 touches the ellipse iff Hyperbola : Equation of tangent at (x1 , y1 ) : Equation of normal at (x1 , y1 ) : 2. x + my + n = 0 touches the hyperbola iff _______________________________ 3. Equation of tangent with slope m : y = mx 6. 4. 5. 6. Equation of chord with (h, k) as mid-point : Locus of mid-point of a system of parallel 7. Equation of chord with (h, k) as mid-point : _________________________________ 8. Locus of mid-point of a system of parallel chords with common slope m : chords with common slope m : b2 y = x a 2m 9. Conjugate diameters : m m1 = _________________________________ 9. b2 a2 10. Auxiliary circle : x2 + y2 = a2 Director circle : _________________________________ hx ky h2 k2 + = + a2 b2 a2 b2 8. Equation of tangent with slope m : _________________________________ x2 + y2 = a 2 + b2 7. Equation of normal at (x1 , y1 ) : _________________________________ b2 + a 2m 2 Director circle : Equation of tangent at (x1 , y1 ) : ________________________________ a2 x b2 y = a 2 - b2 x1 y1 5. b2 = a 2 e 2 - 1 xx1 yy + 1 = 1 a2 b2 4. x2 y2 = 1 a2 b2 1. a 2 2 + b 2 m 2 = n 2 3. 2bt 1 - t2 Conjugate diameters : __________________________________ 10. Auxiliary circle : _________________________________ Pure Maths/Hyperbola/p.4 Example 1. Find the centre and asymptotes of the hyperbola 9x2 - 16y2 + 18x - 32y - 151 = 0 . ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ 2. If the conic x2 + y2 = 1 is tangent to the straight lines 5x - y - 4 = 0 and 5x + 2y + 2 = 0, find the values of and . Hence, or otherwise, find the points of contact of the conic and the straight lines. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ _________________________________________________ y _________________________________________________ _________________________________________________ _________________________________________________ _________________________________________________ x Pure Maths/Hyperbola/p.5 3. Find the equation of the normal, with slope m, to the hyperbola x2 a2 - y2 b2 = 1 y ________________________________________________ _________________________________________________ _________________________________________________ x _________________________________________________ _________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ H.W. : W.43Ex.A. Q.1, 2 Rectangular Hyperbola Consider the rectangular hyperbola xy = a’ , Case 1 : if a’ > 0 , let xy = c2 , y To rotate x, y-axes to X, Y-axes by 45°. X = Y X Y x y xy = c2 45° X 2 - Y 2 = ______________________________________ x 0 = ______________________________________ Case 2 : if a’ < 0 , let xy = - c2 , y Y To rotate x, y-axes to X, Y-axes by - 45°. X = Y x y xy = - c2 X 2 - Y 2 = ______________________________________ = ______________________________________ x 0 - 45° X Pure Maths/Hyperbola/p.6 For xy = c2 , the parametric equations are usually taken as : x = ct y = c t for any non-zero parameter t. Example 1. (a) c Show that the equation of the normal to the hyperbola xy = c2 at the point ct , is t ct 4 - x t 3 + y t - c = 0 . (b) If the normals at the points whose parameters are t1 , t2 , t3 , t4 are concurrent, show that the concurrent point has coordinates given by x = c ( t1 + t2 + t3 + t4 ) and 1 1 1 1 y = c + + + . t2 t3 t4 t1 (a) ________________________________________________________________________ ________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (b) _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Note : If Pi are conormal points on the rectangular hyperbola xy = c2 with parameters ti (I = 1, 2, 3, 4), then t1 t2 t3 t4 = -1 . 2. The parametric coordinates of any point P on the rectangular hyperbola xy = c2 are ct , c . t Four points P1 , P2 , P3 and P4 on the hyperbola have parameters t1 , t2 , t3 and t4 respectively. (a) Prove that P1 , P2 , P3 and P4 lie on a circle if and only if t1 t2 t3 t4 = 1 . (b) State the relationship between the hyperbola and the circle through the points P1, P3 and P4 if t12 t3 t4 = 1 . (c) Given a point P, with parameter t and t2 1, on the hyperbola. Show that there exist two circles which touch the hyperbola at P and again at a second point. If the second point of contact are Q for one circle and R for the other, show that QR passes through the centre of the hyperbola and PQ is perpendicular to PR. Pure Maths/Hyperbola/p.7 Hence, give a sketch of points P, Q, R and the hyperbola. (a) ________________________________________________________________________ ________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (b) _________________________________________________________________________ (c) _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________ y ________________________________________________ _________________________________________________ x _________________________________________________ Classwork 1 Let P t , be a point on the rectangular hyperbola xy = 1. The normal at P meets the hyperbola again t at the point Q. (a) Find the values of t if the tangents at P and Q are parallel. (b) If the tangents at P and Q are not parallel, find the point of intersection of these two tangents, T. Hence, find the equation of the locus of the point T. (a) ________________________________________________________________________ ________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Pure Maths/Hyperbola/p.8 _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (b) _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ W.43 Ex.B Q.6, 9 H.W. W.43 Ex.B Q.1, 2, 7, 8, 10 END