Question

Let S be 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 the $\Delta OPS$ ($O$ is the origin) is ___.

Answer 1

Parametric coordinates to $y^2=4ax$ are $(a{t}^2,2at).$

$P(2t^2,4t)\text{should lie on}{x}^2+y^2-2x-4y=0\Rightarrow 4t^2+16t^2-4t^2-16t=0\Rightarrow 4t^2+12t^2-16t=0\Rightarrow 4t(t^3+3t-4)=0\Rightarrow 4t(t-1)(t^2+t+4)=0\therefore t=0,1\Rightarrow P(2,4)$
Thus, area of $\Delta OPS=\frac{1}{2}\cdot OS\times PQ=\frac{1}{2}\cdot \left(2\right)\cdot \left(4\right)=4$