SATHEE: Parabola 1 Question 8

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) $x = - \frac{a}{2}$

(d) $x = \frac{a}{2}$

Show Answer

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

$\begin{matrix} 4 k^{2} = 4 a (2 h - a) \\ \Rightarrow 4 k^{2} = 8 a h - 4 a^{2} \Rightarrow k^{2} = 2 a h - a^{2} \end{matrix}$

So, locus of $P (h, k)$ is $y^{2} = 2 a x - a^{2}$

$\Rightarrow y^{2} = 2 a x - \frac{a}{2}$

Its directrix is $x - \frac{a}{2} = - \frac{a}{2} \Rightarrow x = 0$ .

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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 y2=4ax is another parabola with directrix

Answer:

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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