Extensions Page 2b

Page 2b

Using the Standard Equation of an Ellipse to
Determine the Location of the Focal Points

If you know the standard equation of an ellipse centered at the origin, x²/a² + y²/b² = 1, you can calculate the coordinates of the foci.
When a > b, the horizontal axis is longer than the vertical axis of the ellipse. Therefore the focal points or foci are located on the horizontal axis at (c, 0) and (-c, 0) where c = (a² - b²)^½.
See this derivation.
When a < b, the vertical axis is longer than the horizontal axis of the ellipse. Therefore the focal points or foci are located on the vertical axis at (0, c) and (0, -c) where c = (b² - a²)^½.
See this derivation.
Let's look at a few examples that show how to find the focal points of an ellipse using these formulas.

EXAMPLE 1:
The ellipse in Figure 1 is defined by the equation x²/ 6 + y²/ 9 = 1.
What are the coordinates of its focal points? Figure 1

Since a < b, you know that the vertical axis is longer than the horizontal axis of the ellipse. This means that the focal points will lie on the vertical axis at (0, c) and (0, -c).
Therefore, we calculate the focal points of the ellipse using c = (b² - a²)^½.
This gives: c = (9 - 6)^½
c = 3^½
c = Ö3 Figure 2

Thus the coordinates of focal points are (0, Ö3) and (0, -Ö3).

EXAMPLE 1:
The ellipse in Figure 1 is defined by the equation x²/ 6 + y²/ 9 = 1. What are the coordinates of its focal points?	Figure 1
Since a < b, you know that the vertical axis is longer than the horizontal axis of the ellipse. This means that the focal points will lie on the vertical axis at (0, c) and (0, -c).
Therefore, we calculate the focal points of the ellipse using c = (b² - a²)^½. This gives: c = (9 - 6)^½ c = 3^½ c = Ö3	Figure 2
Thus the coordinates of focal points are (0, Ö3) and (0, -Ö3).

EXAMPLE 2:
Figure 3
An ellipse is defined by the equation x²/ 49 + y²/ 24 = 1.
What are the coordinates of its foci?
Since a > b, you know that the horizontal axis of the ellipse is longer than the vertical axis. Thus, the focal points will lie on the horizontal axis at (c, 0) and (-c, 0).
Figure 4
Therefore, we calculate the focal points of the ellipse using c = (a² - b²)^½.
This gives: c = (49 - 24)^½
c = 25^½
c = 5
Thus the coordinates of the foci are (5, 0) and (-5, 0).

EXAMPLE 2:
Figure 3	An ellipse is defined by the equation x²/ 49 + y²/ 24 = 1. What are the coordinates of its foci?
Since a > b, you know that the horizontal axis of the ellipse is longer than the vertical axis. Thus, the focal points will lie on the horizontal axis at (c, 0) and (-c, 0).
Figure 4	Therefore, we calculate the focal points of the ellipse using c = (a² - b²)^½. This gives: c = (49 - 24)^½ c = 25^½ c = 5
Thus the coordinates of the foci are (5, 0) and (-5, 0).

Answer the questions in the following question box to practice finding the focal points or foci of an ellipse using the standard equation x²/a² + y²/b² = 1.

Notice that in the above examples and questions you worked with ellipses centered at the origin.
How would you find the focal points of an ellipse centred at (h, k)?