Ellipses

An ellipse is simply an oval: a circle that's been stretched wider than it is tall, or taller than it is wide. Planet orbits are ellipses. The equation looks like a lot of fractions, but it has just four pieces: $h$ and $k$ (where the center is), $a$ (how far it reaches left and right) and $b$ (how far it reaches up and down). If you know the circle's equation, you already know most of this one. It's a small step.

Connecting to the circle. Divide the circle equation $(x - h)^2 + (y - k)^2 = r^2$ by $r^2$ and you get $\frac{(x - h)^2}{r^2} + \frac{(y - k)^2}{r^2} = 1$. An ellipse is the same thing, except the two directions are allowed different numbers under them. That's the whole difference.

Reading the graph. The center is $(h, k)$. To find the edges, notice that when $y = k$ the second fraction is zero, so $(x - h)^2 = a^2$ and $x = h \pm a$. The same argument the other way gives the top and bottom, $(h, k \pm b)$. If $a = b$ it's just a circle of radius $a$.

Getting to standard form. The right side has to be 1. For $4x^2 + 9y^2 = 36$, divide by 36: $\frac{x^2}{9} + \frac{y^2}{4} = 1$, so $a = 3$ and $b = 2$. If there are linear terms like $-4x$, complete the square in each variable first, just as you did for circles.

A common mistake. It's tempting to read $a$ and $b$ straight off the denominators, so that $\frac{x^2}{9}$ means $a = 9$. The denominators are $a^2$ and $b^2$, so here $a = 3$.

Planets orbit the sun in ellipses (Kepler's first law). Whispering galleries and the lithotripters that break up kidney stones use an ellipse's reflective property.