Question

The radius of the circle $r=a\cos \theta +b\sin \theta$ is

• A. $\frac{\sqrt{a^2+b^2}}{2}$
• B. $\sqrt{a^2+b^2}$
• C. $\frac{a^2+b^2}{2}$
• D. $a^2+b^2$

Answer 1

The correct option is A $\frac{\sqrt{a^2+b^2}}{2}$

Put $\cos \theta =\frac{x}{r}$ & $\sin \theta =\frac{y}{r}$ in $r=a\cos \theta +b\sin \theta$
where, $r=\sqrt{x^2+y^2}$
$r=\frac{ax}{r}+\frac{by}{r}$
$\Rightarrow r^2=ax+by$
$\Rightarrow x^2+y^2-ax-by=0$
Comparing with $x^2+y^2+2gx+2fy+c=0$, we get $g=-a/2$ & $f=-b/2$ and $c=0$
Therefore, radius $=\sqrt{g^2+f^2-c}=\frac{\sqrt{a^2+b^2}}{2}$