Question

How do you convert `r(2 - cos theta) = 2` into a rectangular equation?

Answer 1

`3 x^2+4 y^2-4 x -4 = 0`

Explanation:

`2 r = r cos theta + 2=x+2`.

So, `4 r^2=(x+2)^2`.
`4 (x^2+y^2)=x^2+4x+4`
`3 x^2+4 y^2-4 x -4 = 0`

The given equation is `1/r=1-(1/2) cos theta`
This represents an ellipse with eccentricity `e = 1/2` and semi-latus rectum l = 1. Semi-major axis a = `l/(1-e^2)=4/3`.
The negative sign indicates that the initial line is reversed, from pole (a focus) to the center of the ellipse.