Question

If $x=2+3\cos \theta$ and $y=1-3\sin \theta$ represent a circle, then the centre and radius is

• A. $(2,1),9$
• B. $(2,1),3$
• C. $(1,2),\frac{1}{3}$
• D. $(-2,-1),3$

Answer 1

The correct option is B $(2,1),3$

$x=2+3\cos \theta$
$\Rightarrow (x-2{)}^2=(3\cos \theta {)}^2\cdots \left(1\right)$
$y=1-3\sin \theta$
$\Rightarrow (y-1{)}^2=(3\sin \theta {)}^2\cdots \left(2\right)$
Adding $\left(1\right)$ and $\left(2\right),$ we get
$(x-2{)}^2+(y-1{)}^2=3^2$

Comparing with $(x-x_0{)}^2+(y-y_0{)}^2=r^2,$ we get
centre, $(x_0,y_0)=(2,1)$
and radius, $r=3$