Question

If the chord $x\cos \alpha +y\sin \alpha =p$ of the hyperbola $\frac{x^2}{16}-\frac{y^2}{18}=1$ subtends a right angle at the centre, and the diameter of the circle, concentric with the hyperbola to which the given chord is a tangent is $d$ units, then the value of $\frac{d}{4}$ is

Answer 1

Equation of hyperbola is $\frac{x^2}{16}-\frac{y^2}{18}=1$
$\text{or}9x^2-8y^2-144=0$
Homogenizing hyperbola equation using
$\frac{x\cos \alpha +y\sin \alpha }{p}=1$
we get
$9x^2-8y^2-144{\left(\begin{array}{c}\frac{x\cos \alpha +y\sin \alpha }{p}\end{array}\right)}^2=0$
Since these lines are perpendicular to each other
$\therefore 9p^2-8p^2-144({\cos }^2\alpha +{\sin }^2\alpha )=0$
$\Rightarrow p^2=144\text{or}p=\pm 12$
$\Rightarrow$ radius=$\frac{|p|}{\sqrt{co{s}^2\alpha +{\sin }^2\alpha }}=12$
$\therefore \text{radius of the circle}=12$ unit
$\text{and, diameter of the circle}\left(d\right)=24$ unit
$\therefore \frac{d}{4}=6$