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)#
