Question

Consider the circle $x^2+y^2=9$ and the parabola $y^2=8x$. They intersect at $P$ and $Q$ in the first and fourth quadrants, respectively. Tangents to the circle at $P$ and $Q$ intersect the x-axis at $R$ and tangent to the parabola at $P$ and $Q$ intersect the x-axis at $S$.
The ratio of the areas of the triangle $PQS$ and $PQR$ is

• A. $1:\sqrt{2}$
• B. $1:2$
• C. $1:4$
• D. $1:8$

Answer 1

The correct option is C $1:4$

Coordinates of P and Q are $(1,2\sqrt{2})$ and $(1,-2\sqrt{2})$

Area of $△PQR=\frac{1}{2}\times 4\sqrt{2}\times 8=16\sqrt{2}$

Area of $△PQS=\frac{1}{2}\times 4\sqrt{2}\times 2=4\sqrt{2}$

Ration of area of triangle PQS and PQR $=\frac{4\sqrt{2}}{16\sqrt{2}}$

$=\frac{1}{4}$

Therefore ratio is 1:4.