Question

The length of the latus rectum of the parabola $x=a{y}^2+by+c$ is $\frac{k}{a}$. Find $k$

Answer 1

$x=a(y^2+\frac{b}{a}y+\frac{b^2}{4a^2})-\frac{b^2}{4a^2}+c$
$\Rightarrow a{(y+\frac{b}{2a})}^2=x+\frac{b^2}{4a}-c$
$\Rightarrow {(y+\frac{b}{2a})}^2=\frac{1}{a}(x+\frac{b^2-4ac}{4a})$
$\therefore$ the latus rectum $=\frac{1}{a}$