$x² + y² +6y = 7$ to polar coordinates

Question

How do i come from $M1$ to the polar coordinates?

$$M1 \qquad x^2 + y^2 +6y = 7$$

I started with:

$$r^2 * (\cos^2\varphi + \sin^2\varphi)+ 6r\sin\varphi = 7$$ because: $$(\cos^2\varphi + \sin^2\varphi) = 1 $$ means that:

$$r^2 + 6r\sin\varphi = 7$$

But how do I get $r$ on one side? And what are the possible values of $\varphi$? Thanks for your help!

Answer 1

Solve the quadratic $r^2-6r\sin\varphi-7=0$ $$ r=-3\sin\varphi+\sqrt{7+9\sin^2\varphi} $$ Why only $+$? Because that equation has only one positive root, because of the negative constant term.

Answer 2

Hint: Your equation is equivalent to $x^2+(y+3)^2=16$, which is a circle of radius $4$ and center $(0,-3)$