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06%20-%20Hyperbola

# 06%20-%20Hyperbola - PROBLEMS(1 06 HYPERBOLA x2 If a...

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PROBLEMS 06 - HYPERBOLA Page 1 ( 1 ) If a tangent to the hyperbola 1 b y a x 2 2 2 2 = - intersects the major axis in T and minor axis in T’, then prove that 1 CT' b CT a 2 2 2 2 = - , where C is the centre of the hyperbola. ( 2 ) Show that the angle between two asymptotes of the hyperbola x 2 - 2y 2 = 1 is tan - 1 ( 2 2 ). ( 3 ) Prove that the product of the lengths of the perpendicular line segments from any point on the hyperbola 1 b y a x 2 2 2 2 = - to its asymptotes is 2 2 2 2 b a b a + . ( 4 ) Find the co-ordinates of foci, equations of directrices, eccentricity and length of the latus-rectum for the following hyperbolas: ( i ) 25x 2 - 144y 2 = - 3600, ( ii ) x 2 - y 2 = 16. ± ± ± = ± = = = 8 2, e , 2 2 x ), 0 , 2 4 ( ) ii ( 5 288 , 5 13 e , 13 25 y ), 13 0, ( ) i ( : Ans ( 5 ) If the eccentricities of the hyperbolas 1 b y a x 2 2 2 2 ± = - are e 1 and e 2 respectively, then prove that e 1 - 2 + e 2 - 2 = 1. ( 6 ) Prove that the equation of the chord of the hyperbola 1 b y a x 2 2 2 2 = - joining P ( α ) and Q ( β ) is 2 β α cos 2 β α sin b y 2 β α cos a x - - + + = . If this chord passes through the focus ( ae, 0 ), then prove that e 1 e 1 2 β tan 2 α tan + = - ( 7 ) If θ + φ = 2 α ( constant ), then prove that all the chords of the hyperbola 1 b y a x 2 2 2 2 = - joining the points P ( θ ) and Q ( φ ) pass through a fixed point.

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06%20-%20Hyperbola - PROBLEMS(1 06 HYPERBOLA x2 If a...

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