Chapter 9.2, Problem 90AYU

To determine

To give the reason why the vertical line test used to identify functions in rectangular coordinates does not work for equations expressed in polar coordinates.

Explanation of Solution

Vertical line test in rectangular coordinates is used to get symmetry of an equation or function along vertical y axis, in which we replace x coordinate of any point on the graph, with −x, so that to get its left or right image along y axis.

Now as we have that polar coordinate is represented aswhere r is the distance of the point from the pole of polar axis. Now on applying the same ruling of rectangular coordinates, in case of polar coordinates also, or on replacing r by −r, we find another point on the same terminal line at the same distance r from pole, but it different direction, which means that on such replacement, symmetry lies along the pole, not along the vertical line as it lies in the case of rectangular coordinates.