Match the following equation of parabola

$\begin{matrix} Parabola & parametric equation N) x^{2} = 4 a y & 1) (a t^{2}, 2 a t) E) y^{2} = 4 a x & 2) (- a t, 2 a t) W) y^{2} = - 4 a x & 3) (2 a t, a t^{2}) S) x^{2} = - 4 a y & 4) (2 a t, a t^{2}) \end{matrix}$

N-2, E-3, W-1, S - 4

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N-1, E-3, W-2, S - 4

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N-4, E-1, W-2, S - 3

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N-3, E-1, W-2, S - 4

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Solution

The correct option is D
N-3, E-1, W-2, S - 4

The requirements of a parametric representation are it should be able to represent every point on the curve and it should also satisfy the equation of the curve.

But we usually take $(a t^{2}, 2 a t)$ for $y^{2} = 4 a x .$ In our question, we will just substitute and check if it satisfy any equation.

But we need to remember these parametric representation as it will be used later. After substituting and checking we get it as

$y^{2} = 4 a x \to (a t^{2}, 2 a t)$

$x^{2} = 4 a y \to (2 a t, a t^{2})$

$y^{2} = - 4 a x \to (- a t^{2}, 2 a t)$

$x^{2} = - 4 a y \to (2 a t, - a t^{2})$

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Which of the following points lie on the parabola $x^{2} = 4 a y ?$

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