Question

Find the coordinates of the centre of the circle
$r=A\cos \theta +B\sin \theta .$

Answer 1

$r=Acos\theta +Bsin\theta$ .........(1)

Converting to Cartesian coordinate

$x=rcos\theta$

$y=rsin\theta$

Substituting these in equation (1)

$r=\frac{Ax}{r}+\frac{By}{r}$

$r^2=Ax+By$

But $r^2=x^2+y^2$

$\therefore x^2+y^2=Ax+By$

Then, equation of circle becomes

${(x-\frac{A}{2})}^2+{(y-\frac{B}{2})}^2=\frac{\sqrt{A^2+B^2}}{2}$

Centre at $(\frac{A}{2},\frac{B}{2})$ and radius $\frac{\sqrt{A^2+B^2}}{2}$

In polar form $(r,\theta )$, the center is

$(\frac{\sqrt{A^2+B^2}}{2},\arctan \frac{B}{A})$