12.7 Ellipses: from definition to equation

The point ( $x$ , $y$ ) represents any point on the ellipse. $d1$ is its distance from the first focus, and $d2$ to the second. So the ellipse is defined geometrically by the relationship: $d1+d2=10$ .

To calculate $d1$ and $d2$ , we use the Pythagorean Theorem as always: drop a straight line down from ( $x$ , $y$ ) to create the right triangles. Please verify this result for yourself! You should find that $d1=\sqrt{(x+3{)}^2+y^2}$ and $d2=\sqrt{(x-3{)}^2+y^2}$ . So the equation becomes:

$\sqrt{(x+3{)}^2+y^2}+\sqrt{(x-3{)}^2+y^2}=10$ . This defines our ellipse

The goal now is to simplify it. We did problems like this earlier in the year (radical equations, the “harder” variety that have two radicals). The way you do it is by isolating the square root, and then squaring both sides. In this case, there are two square roots, so we will need to go through that process twice.

…and we’re done! Now, according to the “machinery” of ellipses, what should that equation look like? Horizontal or vertical? Where should the center be? What are $a$ , $b$ , and $c$ ? Does all that match the picture we started with?