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.

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.