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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)##