Question

Points $P(-3,2), Q(9,10)$ and $R(\alpha, 4)$ lie on a circle $C$ with $P R$ as its diameter. The tangents to $C$ at the points $Q$ and $R$ intersect at the point $S$. If $S$ lies on the line $2 x-k y-1$, then $k$ is equal to_____

Answer 1

$ m _{ PQ } \cdot m _{ QR }=-1$
$\Rightarrow \frac{10-2}{9+3} \times \frac{10-4}{9-\alpha}=-1 \Rightarrow \alpha=13 $
$ m _{ OP } \cdot m _{ QS }=-1 \Rightarrow m _{ QS }=-\frac{4}{7}$

Equation of QS
$ y-10=-\frac{4}{7}(x-9) $
$ \Rightarrow 4 x+7 y=106 \ldots .(1)$
$ m_{O R} \cdot m_{R S}=-1 \Rightarrow m_{R S}=-8$
Equation of RS
$y-4=-8(x-13) $
$ \Rightarrow 8 x+y=108...$(2)
Solving eq. (1) & (2)
$ x _1=\frac{25}{2} y _1=8 $
$S \left( x _1, y _1\right) \text { lies on } 2 x - ky =1 $
$ 25-8 k =1$
$ \Rightarrow 8 k =24 $
$ \Rightarrow k =3$