Experiment with the applet below by moving points $D$, $E$, and $F$ around.

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Move slider a in the applet below. What do you notice?

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Experiment with the applet below.

What do you notice about the shape that forms?

What do you wonder?

An ellipse is the set of all points for which the sum of the distances to two fixed points (the foci) is fixed. What are the foci of the ellipse in the experiment above?

How would you derive the equation of an ellipse in the $x-$ and $y-$ coordinate plane? Write down your ideas before reading the next paragraphs.

In order to derive the equation of an ellipse, we will use the following applet.

Note that, by definition, the sum of the distances from point $E=(x,y)$ to the two foci, $(c,0)$ and $(-c,0)$, respectively, is constant. That is, $\sqrt{(x-c)^2+y^2}+\sqrt{(x+c)^2+y^2}=$CONSTANT. Note also that we are assuming $c$ is a positive real number.

Move slider $d$ in the applet above to determine the value of the constant. What is it?

(1) Hint: If you haven't already done this, move slider $d$ so that point $E$ is on the positive $x-$axis. Then $E=(a,0)$, for some positive real number $a$, and the sum of the distances from $E$ to the foci is $(a-c)+(a+c)=2a$. Since the sum is constant, it is always $2a$.

(2) Now move slider $d$ so that point $E$ is on the positive $y-$axis. Then $E=(0,b)$, for some positive real number $b$. What do you notice about the value of $b^2+c^2$? Why? (Think about it before reading the following!)

(3) Finally, since the sum of the distances from $E$ to the two foci is $2a$, we have $\sqrt{(x-c)^2+y^2}+\sqrt{(x+c)^2+y^2}=2a$. The reader is encouraged to use algebra to simplify the above before looking at the next paragraph.

After some algebra, we get: $(a^2-c^2)x^2+a^2y^2=(-c^2+a^2)a^2$. In (2) above, you should have determined that $b^2+c^2=a^2$. Thus, $b^2=a^2-c^2$ and plugging this into the simplified equation of the ellipse gives $b^2x^2+a^2y^2=b^2a^2$. Therefore, $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is the equation of the ellipse illustrated in the applet above.

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