Question

How do you graph polar curves to see the points of intersection of the curves?

Answer 1

If the polar equations are `r = f(theta) and r = g(theta)` or, inversely, `theta = f^(-1)(r) and theta = g^(-1)(r)`. eliminate either r or `theta`, solve and substitute in one of the equations..

Explanation:

Explication:

Find the points of intersection of the cardioid

`r = a( 1 + cos theta )` and the circle r = a.

Eliminate r.

The equation for `theta` at a point of intersection is

`a = a(1+cos theta)`. this is #cos theta = 0 rarr theta = pi/2 and

(3pi)/2#.

The common points are `(a, pi/2) and (a, (3pi)/2)`

For the graph, a = 1. Use `(x, y) = r(cos theta, sin theta)`

graph{(x^2+y^2-(x^2+y^2)^0.5-x)(x^2+y^2-1)=0[-2 4 -1.5 1.5]}

The two parabolas `1 = r (1 + cos theta) and 1 = r(1 - cos theta)`

intersect at `(1, pi/2)` and `(1, 3pi/2)`.
graph{(x+(x^2+y^2)^0.5-1)(-x+(x^2+y^2)^0.5-1)=0[-3 3 -1.5 1.5]}