### 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 $Q(-2,2)$

$\therefore(QR)^{2}=36+36=72$