Question

How do you change, this polar form equation `r(2 + cos theta) = 1` to Cartesian form?

Answer 1

`2sqrt(x^2+y^2) + x - 1` I would leave it here optionally you can write (explanation)...
`5x^2+4y^2 + 4xsqrt(x^2+y^2) - 1 = 0`

Explanation:

1) GIVEN: `r(2+costheta) = 1; r(theta) = 1/(2+cos(theta)`
Transform `r(theta) => f(x,y)` Polar => Cartesian
2) RECIPE: `r=x^2+y^2; y=rsintheta; x=rcostheta; theta = tan^-1(y/x)`
3) Solution:
a) Multiply by r(theta) by r:
`2r+rcostheta = 1`
b) Substitute from 2)
`2sqrt(x^2+y^2) + x = 1` Now if you like you can square both sides or stop here...
`a^2 + 2ax + x^2 = 1; " where " a = 2sqrt(x^2+y^2)`
`4(x^2+y^2) + 4xsqrt(x^2+y^2) + x^2 = 1`
`5x^2+4y^2 + 4xsqrt(x^2+y^2) - 1 = 0`