Question

The parametric form of the ellipse $4{(x+1)}^2+{(y-1)}^2=4$ is

• A. $x=\cos \theta –1,y=2\sin \theta –1$
• B. $x=2\cos \theta –1,y=\sin \theta +1$
• C. $x=\cos \theta –1,y=2\sin \theta +1$
• D. $x=\cos \theta +1,y=2\sin \theta +1$
• E. $x=\cos \theta +1,y=2\sin \theta –1$

Answer 1

The correct option is C

$x=\cos \theta –1,y=2\sin \theta +1$

Step 1: Convert the equation in the standard form

The equation of the given ellipse is,

$4{(x+1)}^2+{(y-1)}^2=4$

$\Rightarrow$ $\frac{4{(x+1)}^2+{(y-1)}^2}{4}=\frac{4}{4}$

$\Rightarrow$ $\frac{{(x+1)}^2}{1}+\frac{{(y-1)}^2}{4}=1$

$\Rightarrow$ $\frac{{(x+1)}^2}{1^2}+\frac{{(y-1)}^2}{2^2}=1$

Step 2: Convert into the parametric form

As we know the parametric equations of an ellipse of the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are,

$x=a\cos \theta$ and $y=b\sin \theta$

So, the parametric equations of the given ellipse are,

$x+1=\left(1\right)\cos \theta$

$\Rightarrow$ $x=\cos \theta -1$

And, $y-1=\left(2\right)\sin \theta$

$\Rightarrow$ $y=2\sin \theta +1$

Thus, the required parametric equations of the ellipse are $x=\cos \theta -1$ and $y=2\sin \theta +1$.

Hence, option (C) is the required option.