Question

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

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

Answer 1

The correct option is B $|b|<\frac{1}{2\sqrt{2}}$

Tangent to $y^2=4x$ in terms of $m$ is
$y=mx+\frac{1}{m}$

Normal to $x^2=4by$ in terms of $m$ is
$y=mx+2b+\frac{b}{m^2}$
If these are same lines, then
$\frac{1}{m}=2b+\frac{b}{m^2}$
$\Rightarrow 2b{m}^2-m+b=0$
For two different tangents, $D>0$
$\Rightarrow 1-8b^2>0$
$\Rightarrow |b|<\frac{1}{2\sqrt{2}}$