Parabola 1 Question 8
8. The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $y^{2}=4 a x$ is another parabola with directrix
(2002, 1M)
(a) $x=-a$
(b) $x=-\frac{a}{2}$
(c) $x=0$
(d) $x=\frac{a}{2}$
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Answer:
Correct Answer: 8. (c)
Solution:
- Let $P(h, k)$ be the mid-point of the line segment joining the focus $(a, 0)$ and a general point $Q(x, y)$ on the parabola. Then,
$$ h=\frac{x+a}{2}, k=\frac{y}{2} \Rightarrow x=2 h-a, y=2 k $$
Put these values of $x$ and $y$ in $y^{2}=4 a x$, we get
$$ \begin{gathered} 4 k^{2}=4 a(2 h-a) \\ \Rightarrow \quad 4 k^{2}=8 a h-4 a^{2} \quad \Rightarrow \quad k^{2}=2 a h-a^{2} \end{gathered} $$
So, locus of $P(h, k)$ is $y^{2}=2 a x-a^{2}$
$$ \Rightarrow \quad y^{2}=2 a \quad x-\frac{a}{2} $$
Its directrix is $x-\frac{a}{2}=-\frac{a}{2} \Rightarrow x=0$.
