Question

If the line $lx+my+n=0$ touches the parabola $y^2=4ax$, prove that $ln=a{m}^2$.

Answer 1

The x-coordinates of the points of intersection of the line $lx+my+n=0$ or $y=-\left(\frac{lx+n}{m}\right)$ and the parabola $y^2=4ax$ are roots of the equation

$[-(\frac{lx+n}{n}){]}^2=4ax$

$l^2{x}^2+2x(ln-2a{m}^2)+n^2=0$

If the line $lx+my+n=0$ touches the parabola $y^2=4ax$, then this equation has equal roots.

Therefore,

$4(ln-2a{m}^2{)}^2-4l^2{n}^2=0$

$-4al{m}^{2}n+4a^2{m}^4=0$

$ln=a{m}^2$