The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y 2=4 ax is another parabola with directrix

Question

The locus of the mid-point of the line segment joining the focus to a moving point on the parabola y 2=4 ax is another parabola with directrix (A) x=-a (B) x=-a / 2 (C) x =0 (D) x=a / 2

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/2701577e755bcdfda2b53ad07b671a4a-.png" /> Let P ( x , y ) be the mid point of ( a , 0) and( at 2, 2 at ) 2 x = a (1+ t 2), y = at ∴ 2 x = a [1+(y 2/a 2)]= a +(y 2/a) ⇒ y 2=2 a ( x -(a/2)) Vertex =((a/2), 0) ; Directrix: x =(a/2)-(a/2)=0 ⇒ x =0

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

$x = - a$

$x = - a /2$

$x = 0$

$x = a /2$

Let $P (x, y)$ be the mid point of $(a, 0)$ and $(a t^{2}, 2 a t)$
$2 x = a (1 + t^{2}), y = a t$
$∴ 2 x = a [1 + \frac{y ^{2}}{a ^{2}}] = a + \frac{y ^{2}}{a} \Rightarrow y^{2} = 2 a (x - \frac{a}{2})$
Vertex $= (\frac{a}{2}, 0);$ Directrix $: x = \frac{a}{2} - \frac{a}{2} = 0 \Rightarrow x = 0$

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

x=−a

x=−a/2

x=0

x=a/2