Question

If the tangents drawn from a point to the parabola $y^2=4ax$ are also normals to the parabola $x^2=4by$, then

• A. $b^2\ge 8a^2$
• B. $b^2\ge 2a^2$
• C. $a^2\ge 9b^2$
• D. $a^2\ge 8b^2$

Answer 1

The correct option is D $a^2\ge 8b^2$

The equation of a tangent to $y^2=4ax$ is
$y=mx+\frac{a}{m}\cdots \left(1\right)$
This is a normal to the parabola $x^2=4by$,
So any normal to the parabola is given by
$y=mx+2b+\frac{b}{m^2}\cdots \left(2\right)$
Comparing equation $\left(1\right)$ and $\left(2\right)$, we get
$\frac{a}{m}=2b+\frac{b}{m^2}\Rightarrow 2b{m}^2-am+b=0$
As $m\in R$, so
$D\ge 0\Rightarrow a^2-8b^2\ge 0\Rightarrow a^2\ge 8b^2$