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 the fourth quadrants, respectively. Tangents to the circle at $P$ and $Q$ intersect the $x-$axis at $R$ and tangents to the parabola at $P$ and $Q$ intersect the $x-$axis at $S.$

The ratio of the areas of $\Delta PQS$ and $\Delta PQR$ is

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

Answer 1

The correct option is C $1:4$

Point of intersection of circle and parabola are $P(1,2\sqrt{2}),Q(1,-2\sqrt{2})$
Equation of tangent to parabola P is $y.2\sqrt{2}=4(x+1)$
This cuts the x-axis at $y=0\Rightarrow S=(-1,0)$
Equation of tangent to circle at P is $x.1+y.2\sqrt{2}=9$
This cuts the x-axis at $R=(9,0)$
$\Rightarrow P\equiv (1,2\sqrt{2}),Q=(1,-2\sqrt{2}),R\equiv \left(9.0\right)$ and $S\equiv (-1,0)$
$\therefore \frac{\left(△PQS\right)}{\left(△PQR\right)}=\frac{1}{4}$