What do ##a## and ##b## represent in the standard form of the equation for an ellipse?

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)##