How Do You Find the Equation of an Ellipse Given Its Center, Vertex, and Focus?

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Homework Statement

Center is at (4, -1)
Vertex is at (4, -5)
Focus is at (4, -3.5)

Find the equation of the ellipse.

Homework Equations

horizontal ellipse: ((x-h)^2)/(a^2)) + ((y-k)^2)/(b^2)) = 1
Vertical ellipse: ((y-k)^2)/(a^2)) + ((x-h)^2)/(b^2)) = 1
c^2 = a^2 - b^2

The Attempt at a Solution

The distance between the vertex and the center is 4, so a = 4.

Based on the focus, I got:

-3.5 = 1 + c
c = -2.5

then I did c^2 = a^2 - b^2
to get 6.25 = 16 - b^2
b^2 = 9.75

I put in a^2 (16) and b^2 (9.75) into the equation
Vertical ellipse: ((y-k)^2)/(a^2)) + ((x-h)^2)/(b^2)) = 1
to get: ((y+1)^2)/16) + ((x-4)^2)/9.75) = 1

However, the correct answer is: (4(x-4)^2)/39) + (((y+1)^2)/16). I can't seem to figure out how to arrive at the correct answer. Help is greatly appreciated.

 

[Dick]

4/39=1/9.75.