Question

Tangents are drawn from the points on the parabola $y^2=-8(x+4)$ to the parabola $y^2=4x.$ Then the locus of mid-point of chord of contact of $y^2=4x$ is

• A. $5y^2=8(x+4)$
• B. $5y^2=-4(x-4)$
• C. $5y^2=8(x-4)$
• D. $5y^2=-4(x+4)$

Answer 1

The correct option is C $5y^2=8(x-4)$

Let $(x_1,y_1)$ be point on $y^2=-8(x+4).$
Then equation of chord of contact is $T=0$
i.e., $2x-y_{1}y+2x_1=0\cdots \left(1\right)$

Let $P(h,k)$ be its mid-point.
Then its equation will be
$T=S_1$
$2x-ky+2h=k^2-4h$
$\Rightarrow 2x-ky+k^2-2h=0\cdots \left(2\right)$

Comparing $\left(1\right)$ and $\left(2\right),$ we get
$k=y_1,2x_1=k^2-2h$
So, $k^2=-4(k^2-2h+8)$
$\Rightarrow k^2=\frac{8}{5}(h-4)$
Hence, locus of mid-point of chord of contact is $5y^2=8(x-4)$