Problem 85AE

Equation of an ellipse Consider an ellipse to be the set of points in a plane whose distances from two fixed points have a constant sum 2a. Derive the equation of an ellipse. Assume the two fixed points are on the x-axis equidistant from the origin.

Solution 85AE

Step 1:

In this problem we have to derive the equation of ellipse by using the definition that the sum of the distances between two fixed points is .

Given : The two fixed points are on the x-axis equidistant from the origin.

the major axis is horizontal.

We know that,

An ellipse is "the set of all points in a plane such that the sum of the distances from two fixed points (foci) is constant".

When the major axis is horizontal, the foci are at (-c,0) and at (0,c).

Therefore the sum of the distances to any point (x,y) on the ellipse from the two foci (c,0) and (-c,0) is a 2a.

If we let d1 and d2 bet the distances from the foci to the point, then d1 + d2 = 2a.

Let d1 be the distance from the focus at (-c,0) to the point at (x,y). Since this is the distance between two points, we'll need to use the distance formula.

Similarly, d2 will involve the distance formula and will be the distance from the focus at the (c,0) to the point at (x,y).