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QUESTION

# 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)