Question

Eliminate theta when x = 2+ 3cos theta and y = 5+ 3 sin theta

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{x=2+3\,cos\,\theta}$

$\mathsf{y=5+3\,sin\,\theta}$

$\underline{\textbf{To find:}}$

$\textsf{Eliminate}\;\mathsf{\theta}\;\textsf{from the given equations}$

$\underline{\textbf{Solution:}}$  

$\mathsf{x=2+3\,cos\,\theta\,\implies\,\dfrac{x-2}{3}=cos\,\theta}$---------(1)

$\mathsf{y=5+3\,sin\,\theta\,\implies\,\dfrac{y-5}{3}=sin\,\theta}$----------------(2)

$\textsf{Squaring and adding equations (1) and (2)}$

$\mathsf{\dfrac{(x-2)^2}{9}+\dfrac{(y-5)^2}{9}=cos\,^2\,\theta+sin\,^2\,\theta}$

$\mathsf{\dfrac{(x-2)^2+(y-5)^2}{9}=1}$

$\implies\boxed{\mathsf{(x-2)^2+(y-5)^2=9}}$