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

1 Answer
May 18, 2016

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]}