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)#