Answer:
Step-by-step explanation:
The center is halfway between vertices, at (4, -6).
It is also halfway between foci.
:::::
The vertices are vertically aligned, so the parabola is vertical.
General equation for a vertical ellipse:
(y-k)²/a² + (x-h)²/b² = 1
with
center (h,k)
vertices (h,k±a)
co-vertices (h±b,k)
foci (h,k±c), c² = a²-b²
Apply your data and solve for h, k, a, and b.
center (h,k) = (4, -6)
h = 4
k = -6
vertices (4,-6±a) = (4,-6±9)
a = 9
foci (4,-6±c) = (4,-6±5√2)
b² = a² - c² = 9² - (5√2)² = 31
b = √31
The equation becomes
(y+6)²/81 + (x-4)²/31 = 1
:::::
length of major axis = 2a = 18
length of minor axis = 2b = 2√31
