The equation of the ellipse in standard form is:
[tex]\frac{(x+9)^{2}}{81}+\frac{(y-3)^{2}}{65}=1[/tex]
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 [tex]\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1[/tex]
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
∵ [tex]\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1[/tex]
- Substitute the values in the equation
∴ [tex]\frac{(x+9)^{2}}{81}+\frac{(y-3)^{2}}{65}=1[/tex]
The equation of the ellipse in standard form is:
[tex]\frac{(x+9)^{2}}{81}+\frac{(y-3)^{2}}{65}=1[/tex]
Learn more:
You can learn more about equations of conics equations in brainly.com/question/3333996
brainly.com/question/9390381
#LearnwithBrainly
