More with Polar Curves

Key Questions

  • How do you recognize equivalent polar curves from their equations?

    The easiest way is to input the equations into a graphing calculator. Look at the graph to see if both equations traced the same curve OR look at the table to see if for any given x value, the y-values of each equation are equivalent.

    Patricia M. · 2 · Dec 13 2014
  • How do you graph polar curves to see the points of intersection of the curves?

    Answer:

    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]}

    A. S. Adikesavan · 1 · May 18 2016
  • If two polar curves have different equations, can they be the same curve?

    Yes, they can; for example,

    #r=cos theta#, which looks like:

    enter image source here

    #r=sin(theta-{3pi}/2)#, which looks like:

    enter image source here

    I hope that this was helpful.

    Wataru · 1 · Nov 13 2014
  • Will polar curves always intersect?

    No. Two curves need not intersect.

    Every curve can be expressed in either polar or rectangular form. Some are simpler in one form than the other, but there are not two classes (or families) of curves.

    The curves #x^2+y^2=1# and #x^2+y^2=9# are concentric circles with unequal radii. They do not intersect.

    In polar form, these are the curves #r=1# and #r=3#. (And, of course, they still do not intersect.)

    Jim H · 1 · Mar 24 2015

Questions

The Polar System