Answer:
The values are a = 5 , b = 3 , h = 0 , k = 3
The equation is x²/9 + (y - 3)²/25 = 1
Explanation:
* Lets revise the standard equation of the ellipse
- The standard form of the equation of an ellipse with center (h , k)
and major axis parallel to y-axis is (x - h)²/b² + (y - k)²/a² = 1 , where
-The length of the major axis is 2a
- The coordinates of the vertices are (h , k ± a)
- The length of the minor axis is 2b
- The coordinates of the co-vertices are (h ± b , k)
- The coordinates of the foci are (h , k ± c), where c² = a² - b²
* Now lets solve the problem
∵ The vertices of the ellipse along the major axis are (0 , 8) , (0 , -2)
∴ The major axis is the y-axis
∴ The vertices are (h , k + a) and (h , k - a)
∴ h = 0
∴ k + a = 8 ⇒ (1)
∴ k - a = -2 ⇒ (2)
∵ The foci of it located at (0 , 7) , (0 , -1)
∵ The coordinates of the foci are (h , k + c) and (h , k - c)
∴ h = 0
∴ k + c = 7 ⇒ (3)
∴ k - c = -1 ⇒ (4)
- To find k and a add equations (1) and (2)
∴ (k + k) + (a + - a) = (8 + -2)
∴ 2k = 6 ⇒ divide both sides by 2
∴ k = 3
- Substitute the value of k in equation (1) or (2) to find a
∴ 3 + a = 8 ⇒ subtract 3 from both sides
∴ a = 5
- To find the value of c substitute the value of k in equation (3) or (4)
∴ 3 + c = 7 ⇒ subtract 3 from both sides
∴ c = 4
- To find b use the equation c² = a² - b²
∵ a = 5 and c = 4
∴ (4)² = (5)² - a²
∴ 16 = 25 - b² ⇒ subtract 25 from both sides
∴ -9 = -b² ⇒ multiply both sides by -1
∴ b² = 9 ⇒ take √ for both sides
∴ b = 3
* The values are a = 5 , b = 3 , h = 0 , k = 3
* The equation is x²/9 + (y - 3)²/25 = 1