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Question

# The common tangents to the circle x2+y2=2 and the parabola y2=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
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B
6
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C
9
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D
15
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Solution

## The correct option is D 15Let the equation of tangent to the parabola y2=8x be y=mx+2m It also touches the circle x2+y2=2 ∴∣∣∣2m√1+m2∣∣∣=√2⇒m4+m2=2⇒m4+m2−2=0⇒(m2−1)(m2−2)=0⇒m=±1, m2=−2 [rejected m2=−2] So, tangents are y=x+2, y=-x-2. They, intersect at (-2, 0). Equation of chord PQ is -2x=2 ⇒x=−1 Equation of chord RS is 0=4(x-2) ⇒x=2 ∴ Cooordinates of P, Q, R, S are P(-1,1), Q(-1,-1), R(2, 4), S(2, -4) ∴ Area of quadrilateral =(2+9)×32=15 sq units

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