Question

If a variable line $x\cos \alpha +y\sin \alpha =p$, which is a chord of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(b>a)$. Subtends a right angle at the centre of the hyperbola, then it always touches a fixed circle whose radius is

• A. $\frac{ab}{\sqrt{b-2a}}$
• B. $\frac{a}{\sqrt{a-b}}$
• C. $\frac{ab}{\sqrt{b^2-a^2}}$
• D. $\frac{ab}{b\sqrt{b+a}}$

Answer 1

Answer: (C)

$x\cos \alpha +y\sin \alpha =P$
Subtends a right angle at centre i.e. $(0,0)$
Making homogeneous equation of hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
with the help of $x\cos \alpha +y\sin \alpha =P$
and then coefficient of $x^2+$ coefficient of $y^2=0$
$\frac{1}{a^2}-\frac{1}{b^2}=\frac{1}{P^2}$
$\Rightarrow P=\frac{ab}{\sqrt{b^2-a^2}}$
$P$ is also the length of perpendicular from $(0,0)$ to the line
$x\cos \alpha +y\sin \alpha =P$, then the radius of circle
$P=\frac{ab}{\sqrt{b^2-a^2}}$