Question

How do you graph the system of polar equations to solve `r=1+costheta` and `r=1-costheta`?

Answer 1

graph{x^2+y^2-sqrt(x^2+y^2)-x=0 [-10, 10, -5, 5]} Common points are at `(r, theta)=(1, +-pi/2)`

graph{(x^2+y^2+x)^2-x^2-y^2=0 [-10, 10, -5, 5]}

Explanation:

Using the cartesian equivalents `(x^2+y^2+-x)-sqrt(x^2+y^2)=0`

for `r=1+-cos theta`, the graphs are obtained. The graphs are

are symmetrically opposite, about the y-axis .

By this symmetry, the two meet on the vertical axis, given in

halves, by `theta=+-pi/2`, wherein r=1, at both the common points.

Algebraically adding and subtracting `r=1+-cos theta`, the solutions

`(r, theta)=(1, +-pi/2)` can be obtained.we y