I encountered this problem on my Calculus test today and am struggling to figure it out:
Write $r = \sec^2(\theta)$ as a Cartesian equation.
I have tried using all sorts of tricks on it ($x^2 + y^2 = r^2$, $x = r\cos(\theta)$, $y = r\sin(\theta)$, etc.), but have ended up just going in circles and finding i.e. that $y = \frac{dx}{d\theta} = x\tan(\theta)$ repeatedly.
Any help is appreciated. I've been looking all over the Internet (including here) for an answer to this, but they all seem to be double-angle problems ($\sec(2\theta)$), not this particular one.
Best Answer
We have $x=r\cos\theta$ so $\sec^2 \theta=\frac{r^2}{x^2}=\frac{x^2+y^2}{x^2}$.
Thus our equation becomes $$\sqrt{x^2+y^2}=\frac{x^2+y^2}{x^2}.$$ If we wish, this can be simplified.