JEE PYQ: Parabola Question 2
Question 2 - 2021 (24 Feb Shift 1)
The locus of the mid-point of the line segment joining the focus of the parabola $y^2 = 4ax$ to a moving point of the parabola, is another parabola whose directrix is:
(1) $x = a$
(2) $x = 0$
(3) $x = -\frac{a}{2}$
(4) $x = \frac{a}{2}$
Show Answer
Answer: (2)
Solution
Let $P(at^2, 2at)$ be a point on the parabola. Mid-point of $S(a,0)$ and $P$: $h = \frac{at^2+a}{2}$, $k = \frac{2at}{2}$. So $t = \frac{k}{a}$ and $t^2 = \frac{2h-a}{a}$. Locus: $y^2 = a(2x - a) = 2a\left(x - \frac{a}{2}\right)$. Directrix: $x - \frac{a}{2} = -\frac{a}{2} \Rightarrow x = 0$.
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