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)##