Question

How do I convert the equation `x^2 + y^2 + 2x + 5y = 0` into polar form?

I used the substitutes `x = rcos(theta)` and `y = rsin(theta)`, but got `r^2 + 2rcos(theta) + 5rsin(theta) = 0`, and don't think it's the right answer...

Answer 1

`r=-2costheta-5sintheta`

Explanation:

The relation between polar coordinates `(r,theta)` and Cartesian coordinates `(x,y)` is rightly given by

`x=rcostheta`, `y=rsintheta` and hence `x^2+y^2=r^2`

Hence, we can write `x^2+y^2+2x+5y=0`

as `r^2+2rcostheta+5rsintheta=0`

or `r+2costheta+5sintheta=0`

or `r=-2costheta-5sintheta`

See the graph given below. Created using tool at https://www.desmos.com