Question

How do you graph `r(2 + cos theta) = 1`?

Answer 1

This is polar equation`(1/2)/r=1+(1/2)cos theta` of an ellipse referred to a focus as pole and major axis (in the direction of nearer end) as initial line. Major axis = 4/3 and eccentricity = 1/2..

Explanation:

Compare with the standard form `l/r=1+e cos theta`, the rearranged equation .`(1/2)/r=1+(1/2)cos theta`

`e=1/2 and l = a (1-e^2)=1/2`.
`a=1/(2(1-1/4))=2/3`.
O(0, 0) is a focus S.
The other focus S'`(2ae, pi) = (2/3, pi)`
The center C is the midpoint `(1/3, pi)`, in-between.
Put `theta =0, pi` in the equation to get ends of the major axis,
`A(1/3, 0), A'(1, pi)`

Put `r = a =2/3` to get the ends of the minor axis, B(2/3, 2pi/3), B'(2/3, -2pi/3)#.