Question

The locus of the point of intersection of tangents drawn at the extremities of a normal chord to the parabola $y^2=4ax$ is the curve

• A. $y^2(x+2a)-4a^3=0$
• B. $y^2(x+2a)+4a^3=0$
• C. $y^2(x-2a)+4a^3=0$
• D. $y^2(x-2a)-4a^3=0$

Answer 1

The correct option is B $y^2(x+2a)+4a^3=0$

If $(h,k)$ be the point of intersection of tangents then the equation of chord of contact is $ky=2a(x+h)\cdots \left(1\right)$
Equation of normal $y=mx-2am-a{m}^3\cdots \left(2\right)$
Since both the equations are same, so by comparing we get,
$m=\frac{2a}{k},$ and
$\frac{2ah}{k}=-2am-a{m}^3$
$\Rightarrow \frac{2ah}{k}=-2a\left(\frac{2a}{k}\right)-a{\left(\frac{2a}{k}\right)}^3$
$\Rightarrow k^2(h+2a)=-4a^3$
$\therefore$ The required locus is $y^2(x+2a)+4a^3=0$