The polar coordinates describe an ellipse with a major axis along the y-axis and a minor axis along the x-axis. If you look at the equations, you can see that x reaches its maximum value, 7, when cos(theta)=1, which occurs at theta=0 and 2pi. Likewise, x reaches its minimum value, -7, when cos(theta)=-1, which occurs at theta=pi.

You can perform the same analysis for y, using the values for theta for which sin(theta) reaches its maximum and minimum values, to determine the maximum and minimum values for y.

The minor axis has a length of 14, since it stretches from x=-7 to x=7.

You should be able to find the length of the major axis by determining the max and min values for y.

The formula for the area of an ellipse is (pi)(A)(B) where A and B represent the lengths of the SEMI-major and SEMI-minor axes. Note that the SEMI-major axis is half the length of the major axis; the SEMI-minor and minor axes have a similar relationship.

Todd Werner

Director, Mathnasium West LA

Math Tutoring at Mathnasium Math Learning Centers