Raymond J. answered • 01/12/21
Patient with Ability to Explain in Many Ways
For an ellipse, the general equation is (x-h)2/a2+(y-k)2/b2 = 1.
You're given the two vertices and two co-vertices.
Vertices = (h±a, k) and co-vertices = (h, k±b).
part a)
The length of the major axis = 2a = (10-0) = 10, so a = 5.
The length of the minor axis = 2b = (5-1) = 4, so b = 2.
The center = (h,k) so we can deduce h=5 and k=3 from the two vertices given, so we have C(5,3).
From there we get the equation of (x-5)2/52 + (y-3)2/22 = 1 or (x-5)2/25 + (y-3)2/4 = 1
part b)
We're given the center (4,-3) so we know h = 4 and k = -3. From the one vertex given (4, 3/2) we can determine the distance to center to be 3/2-(-3) = 4 1/2. The length of that axis is twice that distance, or 9, so this is the major axis, and it's vertical. Since the major axis is vertical, that means our general equation will be (x-4)2/b2+(y+3)2/a2=1.
The length of the major axis = 2a = 9, so a = 4 1/2.
The length of the minor axis = 2b = 2, so b = 1.
Hence, our equation is (x - 4)2/(1)2 + (y + 3)2/(4.5)2 = 1 or (x-4)2+(y+3)2/20.25 = 1
Note: if the ellipse is vertical (major axis parallel to the y-axis) then the larger of a2 and b2 is under the (y-k)2.
If the ellipse is horizontal (major axis parallel to the x-axis) then the larger of a2 and b2 is under the (x-h)2.
