Question

The centre of the circle $r=3\cos \theta +4\sin \theta$ is

• A. $(5,{\tan }^{-1}\frac{3}{4})$
• B. $(5,{\tan }^{-1}\frac{4}{3})$
• C. $(\frac{5}{2},{\tan }^{-1}\frac{3}{4})$
• D. $(\frac{5}{2},{\tan }^{-1}\frac{4}{3})$

Answer 1

Answer: (D)

$(\frac{5}{2},{\tan }^{-1}\frac{4}{3})$

$r=3\cos \theta +4\sin \theta$
$\Rightarrow r=5(\frac{3}{5}\cos \theta +\frac{4}{5}\sin \theta )$
$\Rightarrow r=5(\cos \varphi \cos \theta +\sin \varphi \sin \theta ),\varphi ={\tan }^{-1}\left(\frac{4}{3}\right)$
$\Rightarrow r=5\cos (\theta -\varphi )$
This equation is a standard form of circle in polar coordinate.
$r=2a\cos (\phi -\Psi )$ have center at $(a,\Psi )$.
So, the center of given circle is at $(\frac{5}{2},{\tan }^{-1}(\frac{4}{3}\left)\right)$.