Question

A tangent to the parabola $y^2+4x=0$ meets the parabola $y^2=8x$ in $P$ and $Q$. Then the locus of the middle point of $PQ$ is

• A. Ellipse
• B. Circle
• C. Parabola
• D. Hyperbola

Answer 1

The correct option is A Ellipse

Let the middle point of $PQ$ be $S(h,k)$, then equation of chord $PQ$ is $T=S_1$
$ky–4(x+h)=k^2–8h$
$\Rightarrow y=\left(\right)$
Since it is tangent to $y^2+4x=0$.

Applying the condition of tangency
$\frac{k^2-4h}{k}=+\frac{1}{\frac{4}{k}}$
$4(k^2-4h)-k^2=0$
$3k^2=16h$
$y^2=\frac{16}{3}x$
i.e., required locus is parabola.