Which of the following represents a parabola?
Which of the following represents a parabola?
- $x=4 \cos t, y=4 \sin t$
- $x^2-2=-2 \cos t, y=\cos ^2\left(\frac{t}{2}\right)$
- $\sqrt{x}=\tan t, \sqrt{y}=\sec t$
- $x=\sqrt{1-\sin t}, y=\sin \left(\frac{t}{2}\right)+\cos \left(\frac{t}{2}\right)$
Solution
From option (a).
$\begin{aligned} & x=4 \cos t, y=4 \sin t \\ & x^2=16 \cos ^2 t, y^2=16 \sin ^2 t\end{aligned}$
So, $x^2+y^2=16$, so this equation of a circle.
From option (b),
$x^2-2=-2 \cos t, y=\cos ^2\left(\frac{t}{2}\right)$
$x^2=2-2 \cos t$ ...(i)
$y=\frac{1+\cos t}{2}$ ...(ii)
From Eq. (ii),
$\cos t=2 y-1$
Now, $x^2=2-2 \cos t$
$\begin{aligned} & x^2=2-2(2 y-1) \Rightarrow x^2=2-4 y+2 \\ & x^2=4-4 y\end{aligned}$
So, this is represent the parabola.
Asked in: AP EAMCET 2022 (05 Jul Shift 1)
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