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

The correct answer is 3.

mPQ​⋅mQR​=−1
⇒9+310−2​×9−α10−4​=−1⇒α=13
mOP​⋅mQS​=−1⇒mQS​=−74​ Equation of QS
y−10=−74​(x−9)
⇒4x+7y=106….(1)
mOR​⋅mRS​=−1⇒mRS​=−8
Equation of RS
y−4=−8(x−13)
⇒8x+y=108...(2)
Solving eq. (1) & (2)
x1​=225​y1​=8
S(x1​,y1​) lies on 2x−ky=1
25−8k=1
⇒8k=24
⇒k=3