Question

eliminate the parameter and write the corresponding rectangular equation whose graph represents the curvex=1+costhetay=1+2sintheta

Answer 1

Given,

The equation of the curve are,

$\begin{gathered} x=1+\cos \theta \\ y=1+2\sin \theta \end{gathered}$

Taking the equation first as,

$\begin{gathered} x-1=\cos \theta \\ \cos \text{ }\theta=x-1 \end{gathered}$

Taking the equation second as,

$\begin{gathered} y-1=2\sin \theta \\ \sin \text{ }\theta=\frac{y-1}{2} \end{gathered}$

We know that,

$\begin{gathered} \sin ^2\theta+\cos ^2\theta=1 \\ \frac{(x-1)^2}{1}+(\frac{y-1^{}}{2})^2=1 \\ \frac{(x-1)^2}{1}+\frac{(y-1)^2^{}}{4}^{}=1 \end{gathered}$

This is the required rectangular equation.

Hence, option c is correct.