Question

The length of the latusrectum of the parabola $x=10y^2+by+c$ is $\frac{1}{k}$, then $k=$

Answer 1

The equation of parabola can be rewritten as
$x-c=10(y^2+\frac{b}{10}y)\Rightarrow 10{(y+\frac{b}{20})}^2=x+\frac{b^2}{400}-c\Rightarrow {(y+\frac{b}{20})}^2=\frac{1}{10}(x+\frac{b^2}{400}-c)$
Hence length of the latus-rectum will be $\frac{1}{10}$
$\therefore k=10$