Consider the circle `x^2 + y^2 = 9` and the parabola `y^2 = 8x`. They intersect at P and Q in first and fourth quadrant respectively. Tangents to the circle at P and Q intersect the x-axis at R and tangents at the parabola at P and Q intersect the x-axis at S.

Options

A. `1:sqrt2`
B. `1 : 2`
C. `1 : 4`
D. `1 : 8`

Answer

C

Explanation

Solving `x^(2)+y^(2)=9" and "y^(2)=8x` together, we obtain the coordinates of P and Q as `(1, 2sqrt2)" and "(1, -2sqrt2)` respectively.
`:. PQ = 4sqrt2`

The lequations of tangents to the circle `x^(2)+y^(2)=9` at P and Q are `x+2sqrt2=9" and " x-2sqrt2y=9` respectively.
These two intersect at R(9, 0)
`:." Area of "DeltaPQR=1/2xxPQxxMR`
`rArr" Area of "DeltaPQR=1/2xx4sqrt2xx8=16sqrt2" sq. units"`.
The equations of tangents to the parabola `y^(2)=8xat` P and Q are `y=sqrt2x+sqrt2" and "y=-sqrt2-sqrt2`. These two intersect at S(-1, 0).
`:." Area of "DeltaPQS=1/2PQxxMS=1/2xx4sqrt2xx2=4sqrt2" sq. units".`
Hence, Area of `DeltaPQS` : Area of `DeltaPQR=4sqrt2 : 16sqrt2=1 : 4`.