Students will learn what the foci of an ellipse are, in terms of distance between each focus and the ellipse. Then students will learn how to derive the focal length formula for an ellipse. Here's a lesson, and here's practice. Finally, students will learn how to use the standard form and focal length formulas to solve word problems. Here's an excellent video.

TODO: Is there a video demonstration of the whispering chamber in The Statuary Hall of the Capitol Building in Washington, DC?

Concude by giving your students this challenge:

Derive a formula for the area of a triangle, when \(b,\) \(\angle C,\) and \(\angle A\) have been given. Here's the solution: Our goal is to use the formula \((1/2)ab\sin C,\) but currently, the value of \(a\) is unknown. Before finding the value of \(a,\) let's find the third angle from the two which are known:

$$B = \pi - A - C \\[1em]$$

Now that \(B\) is known, we can use the law of sines to find the value of \(a{:}\)

$$\begin{align}
& \dfrac{\sin A}{a} = \dfrac{\sin B}{b} \\[1em]
& a\sin B = b\sin A \\[1em]
& a = \dfrac{b\sin A}{\sin B}
\end{align}$$

Next, substitute for \(B,\) then simplify:

$$\begin{align}
& a = \dfrac{b\sin A}{\sin(\pi - A - C)} \\[1em]
& a = \dfrac{b\sin A}{\sin(A - C)}
\end{align}$$

Finally, substitute for \(a,\) in the formula \((1/2)ab\sin C,\) then simplify:

$$\begin{align}
A &= \tfrac{1}{2}\left[\dfrac{b\sin A}{\sin(A - C)}\right]b\sin C \\[1em]
&= \dfrac{b^2 \sin A \sin C}{2\sin(A - C)}
\end{align}$$