An ellipse has vertices along the major axis at (0, 8) and (0, -2). the foci of the ellipse are located at (0, 7) and (0, -1). what are the values of a, b, h, and k, given the equation below?

The values are a = 5 , b = 3 , h = 0 , k = 3

The equation is x²/9 + (y - 3)²/25 = 1

Step-by-step 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

a=5, b=3, h=0, k=3

Step-by-step explanation:

The center of the circle is (0,3) therefore h is 0 and k is 3. If you use a graphing calculator and plot the points given you should find that a=5. Then try to c and use the equation c^2=a^2-b^2 to find b.

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