How would you represent $y=(x-h)^2+k$ in polar coordinates?

Question

I tried using $$x=r\cos(\theta)$$ and $$y=r\sin(\theta)$$ and ended up with $r\sin(\theta) = (r\cos(\theta)-h)^2 + k$ and wasn't sure how to proceed from there.

Answer 1

Set $$x-h=\sqrt{k} \tan(\theta)$$ So, $$x=h+\sqrt{k} \tan(\theta)$$

Now, $$y=k \tan^2(\theta) + k = k\,{\sec^2(\theta)} $$

Finally, $x=h+\sqrt{k} \tan(\theta)$ and $y=k\,{\sec^2(\theta)} $.