Question

How do you convert `r = 12/(4 + 8 sinx)` into cartesian form?

Answer 1

`x^2-3y^2+12y-9=0`

Explanation:

As `x` is used in Cartesian or rectangular corrdinate, I would use `theta` for angle in place of `x`. The relation between polar coordinates `(r,theta)` and Cartesian coordinates `(x,y)` is given by

`x=rcostheta` and `y=rsintheta` i.e. `r^2=x^2+y^2` and `theta=tan^(-1)(y/x)`

Hence `r=12/(4+8sintheta)`

or `4r+8rsintheta=12`

or `4sqrt(x^2+y^2)=12-8y`

or `sqrt(x^2+y^2)=3-2y`

or `x^2+y^2=9+4y^2-12y`

or `x^2-3y^2+12y-9=0`

which isthe equation of hyperbola.
graph{x^2-3y^2+12y-9=0 [-9.75, 10.25, -3, 7]}