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Question: If a variable straight line x cos a + y sin a = p which is chord of the hyperbola \(\frac{x^{2}}{a^{...

If a variable straight line x cos a + y sin a = p which is chord of the hyperbola x2a2−y2b2\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1(b>a) subtend a right angle at the centre of hyperbola, then it always touches a fixed circle whose radius is equal to –

A

aba2+b2\frac{ab}{a^{2} + b^{2}}

B

abb2−a2\frac{ab}{\sqrt{b^{2} - a^{2}}}

C

aba2+b2\frac{ab}{\sqrt{a^{2} + b^{2}}}

D

None

Answer

abb2−a2\frac{ab}{\sqrt{b^{2} - a^{2}}}

Explanation

Solution

x cos a + y sin a = p

x2a2\frac{x^{2}}{a^{2}} – y2b2\frac{y^{2}}{b^{2}} = 1

intersection point A and B

join eqn of OA, OB

x2a2\frac{x^{2}}{a^{2}} – y2b2\frac{y^{2}}{b^{2}} =

x2a2\frac{x^{2}}{a^{2}} – y2b2\frac{y^{2}}{b^{2}} – x2cos⁡2αp2\frac{x^{2}\cos^{2}\alpha}{p^{2}} – y2sin⁡2αp2\frac{y^{2}\sin^{2}\alpha}{p^{2}}

– 2xypcos⁡αsin⁡αp2\frac{2xyp\cos\alpha\sin\alpha}{p^{2}} = 0

Chord subtend a right angle at the centre.

Coefficient of x2 + coefficient of y2 = 0

1a2\frac{1}{a^{2}} – cos⁡2αp2\frac{\cos^{2}\alpha}{p^{2}} – 1b2\frac{1}{b^{2}} – sin⁡2αp2\frac{\sin^{2}\alpha}{p^{2}} = 0

1a2\frac{1}{a^{2}} – 1b2\frac{1}{b^{2}} = 1p2\frac{1}{p^{2}}

b2−a2a2b2\frac{b^{2} - a^{2}}{a^{2}b^{2}} = 1p2\frac{1}{p^{2}} Ž p = abb2−a2\frac{ab}{\sqrt{b^{2} - a^{2}}}

x cos a + y sin a = p touches x2 + y2 = r2 if length of ^ from (0, 0) upon the line = Radius

if pcos⁡2α+sin⁡2α\frac{p}{\sqrt{\cos^{2}\alpha + \sin^{2}\alpha}} = r Ž p = r

line touches a fixed circle with radius abb2−a2\frac{ab}{\sqrt{b^{2} - a^{2}}}