Question

Let $S$ be the focus of the parabola $y ^{2}=8 x$ and let $PQ$ be the common chord of the circle $x^{2}+y^{2}-2 x-4 y=0$ and the given parabola. The area of the $\Delta PQS$ is

• A. 4 sq units
• B. 3 sq units
• C. 2 sq units
• D. 8 sq units

Answer 1

Answer: (A)

4 sq units

Answer 2

Given Parabola:
$\Rightarrow y ^{2}=8 x$
Given Circle: $( x -1)^{2}+( y -2)^{2}=5$
$\Rightarrow$ Take a point on the parabola as $\left(2 t^{2}, 4 t\right) \equiv(x, y)$
Solve equations simultaneously
$\Rightarrow 4 t ^{4}+16 t ^{2}-4 t ^{2}-16 t =0$
$\Rightarrow t ^{4}+3 t ^{2}-4 t =0$
$\Rightarrow t =0,1$
$\Rightarrow$ We get the points as $P(0,0)$ and $Q(2,4)$.
$\Rightarrow$ The distance between $(0,0) \equiv\left( x _{1}, y _{1}\right)$ and $(2,4) \equiv\left( x _{2}, y _{2}\right)$ is given by,
$\Rightarrow$ Distance Formula $=\sqrt{\left( x _{2}- x _{1}\right)^{2}+\left( y _{2}- y _{1}\right)^{2}}$
$\therefore$ Distance $=2 \sqrt{5}$
$\Rightarrow$ Focus of parabola $y^{2}=8 x$ has coordinates $(2,0)$.
$\Rightarrow P QS$ will form a right angled triangle with area $\frac{1}{2} \times 2 \times 4=4$ square units.