Find the locus of a point whose co-ordinates are given by $x = a t^{2}, y = 2 a t$ , where 't' is a parameter

y^{2} = 2 a x

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x^{2} = 4 a y

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y^{2} = 4 a x

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x^{2} = 2 a y

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Solution

The correct option is C $y^{2} = 4 a x$
We have given, $x = a t^{2}$ and $y = 2 a t$

or $t^{2} = \frac{x}{a}$ .......(i) and $t = \frac{y}{2 a}$ .......(ii)

Now squaring (ii) and equating it to (i) we get,

$\frac{y}{4 a^{2}} = \frac{x}{a} \Rightarrow y^{2} = 4 a x$

Hence required locus is given by, $y^{2} = 4 a x$ , which is a parabola.

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