Question

Find the condition on $a$ & $b$ so that the two tangents drawn to the parabola $y^2=4ax$ from a point are normals to the parabola $x^2=4by$

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

Answer 1

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

Equation of tangent of
$y^2=4ax$ in slope form at $(x_1,y_1)$ is
$y_1=m{x}_1+\frac{a}{m}$ ....(1)
Equation of normal at $(2b{t}_1,b{t}_1^2)$ on $x^2=4by$
$x+t_{1}y=2b{t}_1+b{t}_1^3$
It passes through $(x_1,y_1)$
$\therefore x_1+t_1{y}_1=2b{t}_1+b{t}_1^3$ ....(2)
(1) & (2) are same equation so compare
$\frac{1}{t_1}=-\frac{m}{1}=\frac{a}{m(2b{t}_1=b{t}_1^3)}$
$t_{1}m=-1$
$-m^2{t}_1(2b+b{t}_1^2)=a$
$\Rightarrow m(2b+b{t}_1^2)=a$ ....(3)
Put $m=\frac{1}{t_1}$ in equation (3)
$2b+b{t}_1^2=-a{t}_1$
$b{t}_1^2+a{t}_1+2b=0$
$t_1$ will be real if $D>0$
$a^2>8b^2$