Question

The common tangents to the circle $x^2+y^2=2$ and the parabola $y^2=8x$ touch the circle at the points P & Q and the parabola at the points R & S. Then, the area (in sq units) of quadrilateral PQRS is ?

• A. 3
• B. 6
• C. 9
• D. 15

Answer 1

The correct option is D 15

Let the equation of tangent to the parabola $y^2=8x$ be $y=mx+\frac{2}{m}$
It also touches the circle $x^2+y^2=2$
$\therefore \left|\frac{2}{m\sqrt{1+m^2}}\right|=\sqrt{2}\Rightarrow m^4+m^2=2\Rightarrow m^4+m^2-2=0\Rightarrow (m^2-1)(m^2-2)=0\Rightarrow m=\pm 1,m^2=-2[rejected{m}^2=-2]$
So, tangents are y=x+2, y=-x-2.
They, intersect at (-2, 0).



Equation of chord PQ is -2x=2 $\Rightarrow x=-1$
Equation of chord RS is 0=4(x-2) $\Rightarrow x=2$
$\therefore$ Cooordinates of P, Q, R, S are P(-1,1), Q(-1,-1), R(2, 4), S(2, -4)
$\therefore$ Area of quadrilateral =$\frac{(2+9)\times 3}{2}=15squnits$