Ellipses - Example 2: Finding the Equation

In this lesson, we find the equation of an ellipse given it has vertices or x-intercepts of (8,0) and (-8,0) and focal points (5,0) and (-5,0).

The standard form of the ellipse is:

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

And the x-intercepts are given by:

(\pm a,0)

which means that a = 8.

For the term b², it can be expressed as...

b^2 = a^2 - (ae)^2

where a is the length of the semi-major axis, e is the eccentricity, ae is the distance from the centre of the ellipse to a focal point. So in this case, we have...

ae = 5

Thus

b^2 = 8^2 - 5^2 = 64 - 25 = 39

So the equation of the ellipse is...

\frac{x^2}{64} + \frac{y^2}{39} = 1

or...

39x^2 + 64y^2 = 2496

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