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 [2002]

Question

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]

Smart Achievers · Accepted Answer

(3)

If $(h, k)$ is the mid point of line joining focus $(a, 0)$ and $Q (a t^{2}, 2 a t)$ on parabola then $h = \frac{a + a t^{2}}{2}, k = a t$

Eliminating $t$ , we get $2 h = a + a (\frac{k^{2}}{a^{2}})$

$\Rightarrow k^{2} = a (2 h - a)$ $\Rightarrow$ $k^{2} = 2 a (h - \frac{a}{2})$

$∴$ $Locus of (h, k) is$ $y^{2} = 2 a (x - \frac{a}{2}), which is equation of a parabola$

$whose directrix is (x - \frac{a}{2}) = - \frac{a}{2} \Rightarrow x = 0$

Solution

Q.
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]

1 $x = - a$

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

3 $x = 0$

4 $x = \frac{a}{2}$

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Solution

Q. 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 [2002]

1 x=-a

2 x=-a2

3 x=0

4 x=a2