Question

The center of an ellipse is (−9,3) . One focus is (−13,3) . The major axis is 18 units long. What is the equation of the ellipse in standard form? Enter your answer by filling in the boxes.

Answer 1

The equation of the ellipse in standard form is:

$\frac{(x+9)^{2}}{81}+\frac{(y-3)^{2}}{65}=1$

Step-by-step explanation:

The standard form of the equation of an ellipse with center (h , k)

and major axis parallel to x-axis is $\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1$  

1. The length of the major axis is 2a  

2. The coordinates of the vertices are (h ± a , k )  

3. The coordinates of the foci are (h ± c , k), where c² = a² - b²  

∵ The center of the ellipse is (-9 , 3)

∴ h = -9 , k = 3

∵ One focus is (-13 , 3)

∴ Major axis parallel to x-axis

∵ The length of the major axis = 2a

∵ The length of the major axis = 18

∴ 2a = 18

- Divide both sides by 2

∴ a = 9

∵ The coordinates of the foci are (h ± c , k)

∵ One focus is (-13 , 3)

∴ h ± c = -13

∵ h = -9

∴ -9 - c = -13

- Add 9 to both sides

∴ -c = -4

- Divide both sides by -1

∴ c = 4

∵ c² = a² - b²

∵ c = 4 and a = 9

∴ (4)² = (9)² - b²

∴ 16 = 81 - b²

- Subtract 81 from both sides

∴ -65 = -b²

- Divide both sides by -1

∴ b² = 65

- Substitute the values of a² and b² in the equation of the ellipse

∵ a² = (9)² = 81

∵ b² = 65

∵ h = -9 and k = 3

∵ $\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1$  

- Substitute the values in the equation

∴ $\frac{(x+9)^{2}}{81}+\frac{(y-3)^{2}}{65}=1$

The equation of the ellipse in standard form is:

$\frac{(x+9)^{2}}{81}+\frac{(y-3)^{2}}{65}=1$

Learn more:

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