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Example 7 of finding the equation of an ellipse

Question:

The foci of an ellipse are F1(-4, 0) and F2(4, 0). The point P(8, 6) lies on the ellipse. Find the equation of the ellipse.

How to find the equation of an ellipse based on foci of the ellipse and a point on the ellipse?

Solution:

Because focus F1 and focus F2 lies in the x-axis, so, the standard equation of the ellipse is x2/a2 + y2/b2 = 1, in which a > b > 0.

Use the definition of an ellipse, |PF1| + |PF2| = 2a

2a = |PF1| + |PF2|
= square root of [(8 + 4)2 + 62] + square root of [(8 - 4)2 + 62]
= square root of (144 + 36) + square root of (16 + 36)
= square root of 180 + square root of 52
= 13.4 + 7.2 = 20.6
a = 10.3
a2 = 106

Because the coordinate of the focus related to the constant c, F1 has the coordinate (-c, 0), F2 has the coordinate (c, 0), so, c = 4.

b2 = a2 - c2
= 106 - 16 = 90

So, the equation of the ellipse is: x2/106 + y2/90 = 1. Please watch the video for more details.