Question

Tangents are drawn at the end points of a normal chord of the parabola $y^2=4ax$ . The locus of their point of intersection is

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

Answer 1

Answer: (D)

Let, $P\left(a{t}_1^2,2a{t}_1\right)$ and $Q\left(a{t}_2^2,2a{t}_2\right)$ be the end points of a normal chord of the parabola $y^2=4ax$ such that $PQ$ is normal at $P.$ Then,
$t_2=-t_1-\frac{2}{t_1}$ …(i)
Let, $\left(h,k\right)$ be the point of intersection of tangents to the parabola at $P$ and $Q.$ Then,
$h=a{t}_1{t}_2$ and $k=a\left(t_1+t_2\right)$
$\Rightarrow h=a{t}_1\left(-t_1-\frac{2}{t_1}\right)$ and $k=\frac{-2a}{t_1}.$ [Using (i)]
$\Rightarrow h=-a\left(t_1^2+2\right)$ and $t_1=\frac{-2a}{k}$
$\Rightarrow h=-a\left(\frac{4a^2}{k^2}+2\right)$
$\Rightarrow \left(h+2a\right)k^2=-4a^3$
Hence, the locus of $\left(h,k\right)$ is
$\left(x+2a\right)y^2=-4a^3$
or $\left(x+2a\right)y^2+4a^3=0$