For ellipses, a >= b (when a = b, we have a circle)
a represents half the length of the major axis while b represents half the length of the minor axis.
This means that the endpoints of the ellipse's major axis are a units (horizontally or vertically) from the center (h, k) while the endpoints of the ellipse's minor axis are b units (vertically or horizontally)) from the center.
The ellipse's foci can also be obtained from a and b.
An ellipse's foci are f units (along the major axis) from the ellipse's center
where f^2 = a^2 - b^2
Example 1:
x^2/9 + y^2/25 = 1
a = 5
b = 3
(h, k) = (0, 0)
Since a is under y, the major axis is vertical.
So the endpoints of the major axis are (0, 5) and (0, -5)
while the endpoints of the minor axis are (3, 0) and (-3, 0)
the distance of the ellipse's foci from the center is
f^2 = a^2 - b^2
=> f^2 = 25 - 9
=> f^2 = 16
=> f = 4
Therefore, the ellipse's foci are at (0, 4) and (0, -4)
Example 2:
x^2/289 + y^2/225 = 1
x^2/17^2 + y^2/15^2 = 1
=> a = 17, b = 15
The center (h, k) is still at (0, 0).
Since a is under x this time, the major axis is horizontal.
The endpoints of the ellipse's major axis are at (17, 0) and (-17, 0).
The endpoints of the ellipse's minor axis are at (0, 15) and (0, -15)
The distance of any focus from the center is
f^2 = a^2 - b^2
=> f^2 = 289 - 225
=> f^2 = 64
=> f = 8
Hence, the ellipse's foci are at (8, 0) and (-8, 0)
