Question

The locus of the point $P({\cos }^{2}t,2\sin t),t\in R$ is

• A. $y^2=4x$
• B. $y^2=4x+1$
• C. $y^2+4x=1$
• D. $y^2+4x=4$

Answer 1

The correct option is D $y^2+4x=4$

Let $P(h,k)\equiv ({\cos }^{2}t,2\sin t)$
$\Rightarrow h={\cos }^{2}t......\left(1\right)$
$k=2\sin t$
$\Rightarrow \frac{k}{2}=\sin t$
$\Rightarrow \frac{k^2}{4}={\sin }^{2}t......\left(2\right)$
Adding $\left(1\right)$ and $\left(2\right)$, we get
$\Rightarrow h+\frac{k^2}{4}={\cos }^{2}t+{\sin }^{2}t=1$
$\Rightarrow 4h+k^2=4$
$\therefore$ locus of $(h,k)$ is $4x+y^2=4$