Question

Let $A$ be a point on the $x$-axis. Common tangents are drawn from $A$ to the curves $x^2+y^2=8$ and $y^2=16 x$. If one of these tangents touches the two curves at $Q$ and $R$, then $(Q R)^2$ is equal to

• A. 76
• B. 81
• C. 72
• D. 64

Answer 1

Answer: (C)

$ y = mx +\frac{4}{ m }$
$ \frac{\left|\frac{4}{ m }\right|}{\sqrt{1+ m ^2}}=2 \sqrt{2}$
$ \therefore m =\pm 1 $
$ y =\pm x \pm 4$ Point of contact on parabola
Let $m =1,\left(\frac{ a }{ m ^2}, \frac{2 a }{ m }\right)$
$R (4,8)$
Point of contact on circle $Q(-2,2)$
$\therefore( QR )^2=36+36=72$