Polar Coordinates — Question 3

Question 3 – Solution

Solution

1. Use the polar–Cartesian relationships $$x=r\cos\theta,\qquad y=r\sin\theta,\qquad r^2=x^2+y^2.$$ Equivalent polar coordinates satisfy $$(r,\theta)=(r,\theta+2k\pi)=(-r,\theta+(2k+1)\pi).$$

2. Apply the identity that matches the requested conversion, symmetry test, or intersection, and then check the resulting point or curve in the original polar equation.

3. Since $\cos2\theta=(x^2-y^2)/r^2$, $r^4=9(x^2-y^2)$.

4. Thus $\boxed{(x^2+y^2)^2=9(x^2-y^2)}$.