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

Oct 16, 2014

For ellipses, $a \ge b$ (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 $\left(h , k\right)$ 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:

${x}^{2} / 9 + {y}^{2} / 25 = 1$

$a = 5$
$b = 3$

$\left(h , k\right) = \left(0 , 0\right)$

Since $a$ is under $y$, the major axis is vertical.

So the endpoints of the major axis are $\left(0 , 5\right)$ and $\left(0 , - 5\right)$

while the endpoints of the minor axis are $\left(3 , 0\right)$ and $\left(- 3 , 0\right)$

the distance of the ellipse's foci from the center is

${f}^{2} = {a}^{2} - {b}^{2}$

$\implies {f}^{2} = 25 - 9$
$\implies {f}^{2} = 16$
$\implies f = 4$

Therefore, the ellipse's foci are at $\left(0 , 4\right)$ and $\left(0 , - 4\right)$

Example 2:

${x}^{2} / 289 + {y}^{2} / 225 = 1$

${x}^{2} / {17}^{2} + {y}^{2} / {15}^{2} = 1$

$\implies a = 17 , b = 15$

The center $\left(h , k\right)$ 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 $\left(17 , 0\right)$ and $\left(- 17 , 0\right)$.

The endpoints of the ellipse's minor axis are at $\left(0 , 15\right)$ and $\left(0 , - 15\right)$

The distance of any focus from the center is

${f}^{2} = {a}^{2} - {b}^{2}$
$\implies {f}^{2} = 289 - 225$
$\implies {f}^{2} = 64$
$\implies f = 8$

Hence, the ellipse's foci are at $\left(8 , 0\right)$ and $\left(- 8 , 0\right)$