Convert the equation $(x^2 + y^2)^2 = 2xy$ into polar coordinates.
I tried converting the equation in the image to polar coordinates and got $r= \sqrt{\sin(2\theta)}$.
This answer is wrong, but I do not understand why. Can someone please explain what the correct answer is?
Did you get something similar to this?
For the original equation $$(x^2 + y^2)^2 = 2 x y$$
Let $$\begin {matrix} x = r \cos \theta \\ y = r \sin \theta \\ r^2 = x^2 + y^2 \end {matrix}$$
Plugging all of this in to the original equation we get
$$(r^2)^2 = 2r^2 \sin \theta \cos \theta$$
or
$$r^4 = 2 r^2 \sin \theta \cos \theta$$
Dividing by $r^2$ we have
$$r^2 = 2 \sin \theta \cos \theta$$
However, $2 \sin \theta \cos \theta = \sin 2 \theta$ so we end up with
$$r^2 = \sin 2 \theta$$
We can leave as-is (I think this may be the answer they're expecting) or take square roots...
$$r = \pm \sqrt {\sin 2 \theta}$$