Question

Which of the following represents the parametric form of the parabola $y^2=4ax$?

• A. None of the above.
• B. $x=\frac{1}{4a{t}^2},y=\frac{1}{t}$
• C. $x=\frac{t^2}{4a},y=t$
• D. $x=a{t}^2,y=2at$

Answer 1

Answer: (C), (D)

$x=a{t}^2,y=2at$

$\underset{–}{\text{Option}1}:$ $x=\frac{t^2}{4a},y=t$
$\therefore x=\frac{t^2}{4a}\Rightarrow 4ax=t^2$
$\text{Put value of}y=t\Rightarrow 4ax=y^2$
Hence it is correct.

$\underset{–}{\text{Option}2}:$ $x=a{t}^2,y=2at$
$y^2=4a^2{t}^2\Rightarrow y^2=4a\times a{t}^2$
And we know $a{t}^2=x$
$\Rightarrow y^2=4ax.$
Thus, this also represents the parametric coordinates of parabola.

$\underset{–}{\text{Option}3}:$ $x=\frac{1}{4a{t}^2},y=\frac{1}{t}$
As we know $(0,0)$ will also be the point on the parabola and putting $t=0$ will be not defined. So, this is not possible.
Hence, this does not be representing the parametric coordinate sof the parabola.

Therefore, options (a), (b) are correct answer.