Question

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

• A. $a^2>8b^2$
• B. $b^2>8a^2$
• C. $a^2<8b^2$
• D. None of these

Answer 1

The correct option is A $a^2>8b^2$

The coordinates of any point one the parabola $x^2=4by$ are $(2bt,b{t}^2)$.

For the parabola $x^2=4by,\frac{dy}{dx}=\frac{x}{2b}$

Slope of the normal at $(2bt,b{t}^2)=-\frac{2b}{2bt}=-\frac{1}{t}$

$\therefore$ Equation of normal is $y-b{t}^2=-\frac{1}{t}(x-2bt)\Rightarrow y=-\frac{x}{t}+2b+b{t}^2$

It will touch the parabola $y^2=4ax$ if

$2b+b{t}^2=\frac{a}{-1/t}(\because c=\frac{a}{m})$

$\Rightarrow b{t}^2+at+2b=0$

For distinct real roots, discriminant $>0$

$\Rightarrow a^2-8b^2>0\Rightarrow a^2>8b^2$