Oblique asymptote polar equation

Question

I have the polar equation $r(\theta)=\frac{1}{\theta-\frac{\pi}{4}}$. I can see that it has an oblique asymptote for $\theta \rightarrow\pi/4+$, but what is it in Cartesian form ?

Answer 1

Asymptote means $r\rightarrow\infty$. It gives immediately $\theta=\frac{\pi}{4}$ in polar ie $y=x$ in cartesian gives the direction of the asymptote. See Abel's post for a complete answer.

Answer 2

the asymptote is parallel to the the line $y = x$ not necessarily $y = x.$ i will try to find the asymptote by looking for in the form $$y = x + a $$

in polar coordinates, the asymptote is $$r \sin \theta = r\cos \theta + a\to a=\frac{\sin \theta - \cos \theta}{\theta - \pi/4}= \sqrt 2\frac{\sin (\theta - \pi/4)}{\theta - \pi/4} \rightarrow\sqrt 2 \text{ as } \theta \to \pi/4.$$

so the asymptote is $$y = x + \sqrt 2. $$