Question

How do you convert `r = 1-2 cosθ` into rectangular forms?

Answer 1

`(x^2+y^2- bx)^2 = a^2(x^2+y^2)`
a = 1; b = -2
`(x^2+y^2+ 2x)^2 = (x^2+y^2)`

Explanation:

use the Polar to Cartesian transformation:
`r^2 = x^2 + y^2`
`x = rcostheta`

`r = a + bcostheta`
`r^2 = ar + rbcostheta`
`r^2- rbcostheta =ar`
`x^2+y^2- bx = ar` square the entire thing and write
`(x^2+y^2- bx)^2 = (ar)^2`
`(x^2+y^2- bx)^2 = a^2(x^2+y^2)`

a = 1; b = -2
`(x^2+y^2+ 2x)^2 = (x^2+y^2)`