Question

The radius of the circle with the polar equation $r^2-8r(\sqrt{3}\cos \theta +\sin \theta )+15=0$ is

• A. $8$
• B. $7$
• C. $6$
• D. $5$

Answer 1

The correct option is B $7$

Given polar equation of circle is
$r^2-8(\sqrt{3}\cos \theta +\sin \theta )+15=0$
or $r^2-8(\sqrt{3}r\cos \theta +r\sin \theta )+15=0$
where $r\cos \theta =x$ and $y=r\sin \theta$
It can be rewritten in cartesian form
$x^2+y^2-8\sqrt{3}x+y+15=0$
$\Rightarrow x^2+y^2-8\sqrt{3}x-8y+15=0$
Now, radius $=\sqrt{4{\sqrt{3}}^2+4^2-15}$
$=\sqrt{48+16-15}=7$