Parabola Question 5
Question 5 - 30 January - Shift 2
Let $A$ be a point on the x-axis. Common tangents are drawn from $A$ to the curves $x^{2}+y^{2}=8$ and $y^{2}=$ 16x. If one of these tangents touches the two curves at $Q$ and $R$, then $(QR)^{2}$ is equal to
(1) 64
(2) 76
(3) 81
(4) 72
Show Answer
Answer: (4)
Solution:
Formula: Equation of the Tangent at any Point, Distance between point and line
$y=mx+\frac{4}{m}$
$\frac{|\frac{4}{m}|}{\sqrt{1+m^{2}}}=2 \sqrt{2}$
$ \therefore m= \pm 1$
$y= \pm x \pm 4$.
Point of contact on parabola
Let $m=1,(\frac{a}{m^{2}}, \frac{2 a}{m})$
$R(4,8)$
Point of contact on circle $Q(-2,2)$
$\therefore(QR)^{2}=36+36=72$
