Question

A normal is drawn to the parabola $y^2=4ax$ at the point $(2a,-2\sqrt{2}a)$ then the length of the normal chord, is

• A. $4\sqrt{2}a$
• B. $6\sqrt{2}a$
• C. $4\sqrt{3}a$
• D. $6\sqrt{3}a$

Answer 1

The correct option is D $6\sqrt{3}a$

REF.Image.

Equation of tangent at P:

$y\times (-2\sqrt{2}a)=2a(x+2a)$

$\Rightarrow x+y\sqrt{2}+2a=0$

Equation of normal at P :

$y+2\sqrt{2}a=\sqrt{2}(x-2a)$

$\Rightarrow x\sqrt{2}=4\sqrt{2}a$

Q $(x,\sqrt{4ax})$ lies on the normal

$\Rightarrow y^2\sqrt{2}-4ay=16\sqrt{2}{a}^2$

$\Rightarrow y^2-2\sqrt{2}ay-16a^2=0$

$\Rightarrow y=\frac{2\sqrt{2}a\pm \sqrt{8a^2+64a^2}}{2}=\frac{2\sqrt{2}a+6\sqrt{2}a}{2}=a\sqrt{2}(1+3)$

$\Rightarrow y=4\sqrt{2}a$ and x = 8a

and

$y=-2\sqrt{2}$ and x = 2a

$\therefore$ the co-ordinate of Q are $(8a,4\sqrt{2}a)$

$\therefore$ Length of normal chord = PQ = $\sqrt{36a^2+72a^2}=6\sqrt{3}a$

$\therefore \left(D\right)$ is the correct answer