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Q 6

Expert-verified
Found in: Page 771

Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

Explain why there are infinitely many different ellipses with the same foci.

We know that an ellipse is defined as the set of points in the plane for which the sum of the distances of the foci is a constant. Also there are infinitely many sums which can be obtained in this way. So that there are infinitely many different elapses with the same foci.

See the step by step solution

Step 1. Given Information

We have to explain that why there are infinitely many ellipses with the same foci.

We will use the definition of ellipse to explain this thing.

Step 2. Required explanation

The ellipse is defined as :-

Given two points in a plane, called foci, an ellipse is the set of points in the plane for which the sum of the distances to the two foci is a constant.

Foci are two fixed points.

Let these points are $a,b$.

Then we can choose so many points for which sum of the distances to the two points is constant. That is :-There are so many points that satisfies the condition that the sum of the distances to the two foci is a constant.

That is there are infinitely many sums which can be obtained in this way, there are infinitely many different ellpses with the same foci.

So we can conclude that :-

We know that an ellipse is defined as the set of points in the plane for which the sum of the distances of the foci is a constant. Also there are infinitely many sums which can be obtained in this way. So that there are infinitely many different elapses with the same foci.