Question

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

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

Answer 1

The correct option is A $4$ sq units

The parametric equations of the parabola $y^2=8x$ are $x=2t^2$ and $y=4t$
And the given equation of circle is $x^2+y^2-2x-4y=0$
On putting $x=2t^2$ and $y=4t$ in circle, we get
$4t^4+16t^2-4t^2-16t=0$
$\Rightarrow 4t^2+12t^2-16t=0\Rightarrow t=0,t=1[t^2+t+4\ne 0]$
Thus the coordinates of points of intersection of the circle and the parabola are $Q(0,0)$ and $P(2,4)$.

Clearly these are diametrically opposite points on the circle.

The coordinates of the focus $S$ of the parabola are $(2,0)$ which lies on the circle.
Therefore, area of $△PQS=\frac{1}{2}\times QS\times SP=\frac{1}{2}\times 2\times 4$
$=4$ sq. units