Question

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

• A. $a^2>2b^2$
• B. $a^2<4b^2$
• C. $a^2>8b^2$
• D. $a=2b$

Answer 1

Answer: (C)

$a^2>8b^2$

For $y^2=4ax$, the equation of a tangent can be written as :
$y=mx+\frac{a}{m}$
For $x^2=4by$, the equation of a normal can be written as :
$y=mx+2b+\frac{b}{m^2}$
Comparing the coefficients we get,
$\frac{a}{m}=+2b+\frac{b}{m^2}$
On simplifying we get,
$2b{m}^2-am+b=0$
The roots of this equation must be real and distinct. Hence,
$\Delta >0$
$\Rightarrow a^2-8b^2>0$
$\Rightarrow a^2>8b^2$
Hence,
Option C is correct.