Question

Line $y=2x-p$ cuts the parabola $y=x^2-4x$ at points $A$ and $B$. If $\angle AOB=\frac{\pi }{2}$, where $O$ is the origin, then the value of $p$ is

Answer 1

By homogenization,
$\left(\frac{2x-y}{p}\right)y=x^2-4x\left(\frac{2x-y}{p}\right)$
$\Rightarrow 2xy-y^2=p{x}^2-8x^2+4xy$
$\Rightarrow (8-p)x^2-2xy-y^2=0$
Coefficient of $x^2+$ coefficient of $y^2=0$
$\Rightarrow 8-p-1=0$
$p=7$