Question

If tangent to the parabola $y^2=4x$ is also normal to $x^2=4by$ and $|b|\le \frac{1}{\sqrt{k}}$, then the numerical value of $k$ is

Answer 1

Any tangent to $y^2=4x$ is
$y=mx+\frac{1}{m}$
Any normal to $x^2=4by$ is
$y=mx+2b+\frac{b}{m^2}$,
By equating both the lines, we get
$2b{m}^2-m+b=0$
This equation has real roots, if
$D\ge 0\Rightarrow 1-8b^2\ge 0\Rightarrow |b|\le \frac{1}{\sqrt{8}}\therefore k=8$