Extensions Page 2b.3

Page 2b.3

Equation of an Ellipse Centred at (h, k)

You have already learned that when an ellipse is translated from being centred at the origin to being centred at any point (h, k), the equation of the ellipse changes from x²/a² + y²/b² = 1 to (x - h)²/a² + (y - k)²/b² = 1.
This translation results in the entire ellipse shifting horizontally h units and vertically k units.
Therefore, when a > b, the focal points are located at (c + h, k) and (-c + h, k) where c = (a² - b²)^½.
Similarly, when a < b, the focal points are located at (h, c + k) and (h, -c + k) where c = (b² - a²)^½.
Let's look at a few examples that show how find the focal points of an ellipse using these formulas.

EXAMPLE 1:
An ellipse is defined by the equation (x - 1)²/ 16 + (y - 2)²/ 25 = 1. See Figure 1.
What are the coordinates of its focal points? Figure 1

Since a < b, you know that the vertical axis of the ellipse is longer than the horizontal axis. Also, since the ellipse is centred at (1, 2), you know that the focal points will lie on the vertical line x = 1, with one focal point on either side of the center.
Therefore, we calculate the focal points using c = (b² - a²)^½
This gives: c = (25 - 16)^½
c = (9)^½
c = 3 Figure 2

Thus the coordinates of focal points are (1, 2+3) and (1, 2-3). That is, (1, 5) and (1, -1). See Figure 2.

EXAMPLE 1:
An ellipse is defined by the equation (x - 1)²/ 16 + (y - 2)²/ 25 = 1. See Figure 1. What are the coordinates of its focal points?	Figure 1
Since a < b, you know that the vertical axis of the ellipse is longer than the horizontal axis. Also, since the ellipse is centred at (1, 2), you know that the focal points will lie on the vertical line x = 1, with one focal point on either side of the center.
Therefore, we calculate the focal points using c = (b² - a²)^½ This gives: c = (25 - 16)^½ c = (9)^½ c = 3	Figure 2
Thus the coordinates of focal points are (1, 2+3) and (1, 2-3). That is, (1, 5) and (1, -1). See Figure 2.

EXAMPLE 2:
Figure 3
Consider the ellipse in Figure 3. This ellipse is defined by the equation (x + 3)²/ 24 + (y - 4)²/ 8 = 1.
Determine the coordinates of its foci.
Since a > b, you know that the horizontal axis of the ellipse is longer than the vertical axis. Since the ellipse is centred at (-3, 4) you know that the focal points will lie on the line y = 4, with one focal point on either side of the center.
Figure 4
Therefore, we calculate the focal points using c = (a² - b²)^½. c = (24 - 8)^½
c = 16^½
c = 4
Thus the coordinates of the foci are (-3+4, 4) and (-3-4, 4). That is, the foci are (1, 4) and (-7, 4). See Figure 4.

EXAMPLE 2:
Figure 3	Consider the ellipse in Figure 3. This ellipse is defined by the equation (x + 3)²/ 24 + (y - 4)²/ 8 = 1. Determine the coordinates of its foci.
Since a > b, you know that the horizontal axis of the ellipse is longer than the vertical axis. Since the ellipse is centred at (-3, 4) you know that the focal points will lie on the line y = 4, with one focal point on either side of the center.
Figure 4	Therefore, we calculate the focal points using c = (a² - b²)^½. c = (24 - 8)^½ c = 16^½ c = 4
Thus the coordinates of the foci are (-3+4, 4) and (-3-4, 4). That is, the foci are (1, 4) and (-7, 4). See Figure 4.

Answer the questions in the following question box to practice finding the focal points of an ellipse using the standard equation (x - h)²/ a² + (y - k)²/ b² = 1.