All respective information associated with the ellipse is: h = 4, k = - 2, a = 2, b = 6.
How to derive the information related to an ellipse
In this problem we must derive all information related to an ellipse: vertex coordinates, semiaxis lengths. This can be done by applying algebra properties on general form of the ellipse equation to find vertex form. First, write the entire expression:
36 · x² + 4 · y² - 288 · x + 16 · y + 448 = 0
Second, complete the square:
36 · (x² - 8 · x) + 4 · (y² + 4 · y) + 448 = 0
36 · (x² - 8 · x + 16 - 16) + 4 · (y² + 4 · y + 4 - 4) + 448 = 0
36 · (x² - 8 · x + 16) + 4 · (y² + 4 · y + 4) + 36 · (- 16) + 4 · (- 4) + 448 = 0
36 · (x - 4)² + 4 · (y + 2)² - 144 = 0
36 · (x - 4)² + 4 · (y + 2)² = 144
Third, find the vertex form of the ellipse:
Fourth, extract all respective information:
Vertex: (h, k) = (4, - 2)
a = 2, b = 6