Question

The length of the latus rectum of the parabola $x=10y^2+by+c,$ where $b$ and $c$ are constants, is $\frac{1}{k}.$ Then $k$ is equal to

Answer 1

The equation of parabola can be rewritten as
$x-c=10(y^2+\frac{b}{10}y)\Rightarrow 10{(y+\frac{b}{20})}^2=x+\frac{b^2}{40}-c\Rightarrow {(y+\frac{b}{20})}^2=\frac{1}{10}(x+\frac{b^2}{40}-c)$
Hence, length of the latus rectum $4a$ will be $\frac{1}{10}$
$\therefore k=10$