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

$7$

Solve for the radius of the circle.

$$\begin{gathered} r^2-8r(\sqrt{3}\cos \theta +\sin \theta )+15=0 \\ \Rightarrow r^2-8(\sqrt{3}r\cos \theta +r\sin \theta )+15=0 \end{gathered}$$

where$r\cos \theta =x,r\sin \theta =y$

It can be rewritten in cartesian form as,

$$\begin{gathered} x^2+y^2-8\left(\sqrt{3}x+y\right)+15=0 \\ \Rightarrow x^2+y^2-8\sqrt{3}x-8y+15=0 \end{gathered}$$

Now, radius

$$\begin{gathered} =\sqrt{{\left(4\sqrt{3}\right)}^2+4^2-15}=0 \\ =\sqrt{48+16-15} \\ =7 \end{gathered}$$

Hence, option(B) is the correct answer.