Question

How do you convert `X + Y = 0` to polar form?

Answer 1

`r=0, theta=3pi/4 and theta=7pi/4`

Explanation:

`X=r cos theta and Y = r sin theta`.
So, X + Y = 0 becomes `r(cos theta + sin theta) = 0`.

The solutions are `r = 0 and cos theta + sin theta = 0`..

`So, tan theta = -1`. This gives two solutions in `[0, pi]`, as given in the answer.

Some niceties:

Note that, in polar coordinates, r = 0 gives the pole but `theta` = constant gives the half line from the pole in that direction, sans pole Pole is a point of discontinuity..
I think that I have given justification for giving three polar equations for the whole line represented by X + Y = 0, in rectangular coordinates.

I consider pole r = 0 as only the limit of r, upon reaching the end called pole (origin), along any radial line `theta` = constant. ..