The common tangents to the circle `x^2 + y^2 =2` and the parabola `y^2 = 8x` touch the circle at `P,Q` andthe parabola at `R,S`. Then area of quadrilateral `PQRS` is

Options

A. 3
B. 6
C. 9
D. 15

Answer

D

Explanation

4 Equation of tangent to parabola `y^(2)=8x` having slope m is
`y=mx+(2)/(m)`

It is also tangent to circle `x^(2)+y^(2)=2`
`rArr((2)/(m))/(sqrt(1+m^(2)))=pmsqrt(2)`
`rArrm=pm1`
So, equation of tangent are `y=pm(x+2)`.
Chord of contact `PQ:(-2)x+y(0)=2orx=-1`
Chord of contact `RS:yxx0=4(x-2)orx=2`
Line x=-1 meets the circle at points P(-1,1) and Q-1,-1).
Line x=-2 meets the circle at points R(2,4) and S(2,-4).
Area of trapezium PQRS `=(1)/(2)(PQ+RS)xx("Height")=(1)/(2)(10)xx(3)=15` sq. units