An ellipse and circle with the same area

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Question from Michael, a student:

The area of the ellipse if equal to the area of a circle with radius=40 ft.
Find the values of a and b using appropriate algebraic techniques, not basic math computations.
A=pi ab and satisfy the constraint a+b=100????

Hi Michael,

The area of a circle is π r² where r is the radius and the area of an ellipse is π ab where a and b are the semi-axes lengths. Thus if the circle has radius 40 ft and has the same area as an ellipse then

π 40² = π ab

and thus

1600 = ab

You also have the condition that a + b = 100.

What two numbers a and b have their sum equal to 100 and their product equal to 1600?

Penny

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