Question

The radius of the circle $r=12\cos \theta +5\sin \theta$ is

• A. $\frac{5}{12}$
• B. $\frac{17}{2}$
• C. $\frac{15}{2}$
• D. $\frac{13}{2}$

Answer 1

Answer: (D)

Given equation of circle is
$r=12\cos \theta +5\sin \theta$
Put $\cos \theta =\frac{x}{r}$ and $\sin \theta =\frac{y}{r}$, we get
$r=12\times \frac{x}{r}+5\times \frac{y}{r}$
$\Rightarrow r^2=12x+5y$
$\Rightarrow x^2+y^2=12x+5y$
$\Rightarrow x^2+y^2-12x-5y=0$
$\therefore$ Centre is $\left(6,\frac{5}{2}\right)$.
$\therefore \text{ Radius of circle }=\sqrt{(6{)}^2+{\left(\frac{5}{2}\right)}^2}$
$=\sqrt{36+\frac{25}{4}}=\sqrt{\frac{144+25}{4}}$
$=\sqrt{\frac{169}{4}}=\frac{13}{2}$