Test Index

Parabola Part 1

Section: Mathematics

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Question : 52 of 100

Marks: +1, -0

The locus of the mid-point of the line segment joining the focus of the parabola

y^{2}=4 a x

to a moving point of the parabola, is another parabola whose directrix is

[24 Feb 2021 Shift 1]

$x=a$
$x=-\frac{a}{2}$
$x=0$
$x=\frac{a}{2}$

Solution:

Given, equation of parabola

\Rightarrow y^{2}=4 a x

Focus

=S(a, 0)

Let any point on the parabola be

P(at^{2}, 2 at)

.

and let the mid-point of

P S

be

M(h, k)

.

\;\therefore \;h=\frac{a t^{2}+a}{2} ;\;k=\frac{2 a t+0}{2}

\Rightarrow\;t^{2}=\frac{2 h-a}{a} ; t \;\; =\frac{k}{a}

\;\Rightarrow \;\; t^{2} \;\; =\frac{k^{2}}{a^{2}}

\;\text{ Now, }\;\frac{2 h-a}{a}=\frac{k^{2}}{a^{2}}

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

\therefore \;\text{ Locus of }(h, k) \;\text{ is }\; y^{2}=a(2 x-a)

\;y^{2}=2 a\left(x-\frac{a}{2}\right)

\therefore

The directrix of this parabola is

x-\frac{a}{2}=-\frac{a}{2} \Rightarrow x=0

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