Question

If two different tangents of $y^2=4ax$ are normals to $x^2=4y$, then

• A. $|a|>2\sqrt{2}$
• B. $|a|<2\sqrt{2}$
• C. $|a|>\frac{1}{2\sqrt{2}}$
• D. $|a|<\frac{1}{2\sqrt{2}}$

Answer 1

The correct option is A $|a|>2\sqrt{2}$

The tangent to the parabola $y^2=4ax$ is $y=m_{1}x+\frac{a}{m_1}...\left(1\right)$
The normal to the parabola $x^2=4y$ is $x=m_{2}y-2m_2-(m_2{)}^3...(2)$
comparing equation $\left(1\right)$ and $\left(2\right)$
$m_1=\frac{1}{m_2}$ & $\frac{a}{m_1}=2+(m_2{)}^2$
$\Rightarrow (m_2{)}^2-a{m}_2+2=0$
For two distinct tangents/normals
$D>0$
$a^2-8>0$
$\Rightarrow |a|>2\sqrt{2}$