Find the equation of this ellipse... I'm lost.

[bluebirdsfly]

The perimeter of a triangle is 30, and the points (0,-5) and (0,5) are two fo the vertices. Find the equation of the ellipse of the third vertex.

I'm completely lost; I don't even know where to start... any help?
Btw, this problem is from the Analytic Geometry by Fuller/Tarwater, pg 123, #38.

 

[soroban]

Hello, bluebirdsfly!

I'll solve it quickly.
. . But they are probably expecting to see a lot of algebra . . .

The perimeter of a triangle is 30, and the points (0,-5) and (0,5) are two of the vertices.
Find the equation of the ellipse of the third vertex.

We are expected to know the definition of an ellipse.

\(\displaystyle \text{Given two fixed points }A\text{ and }B\text{, the ellipse is the set of all points }P(x,y)\)
. . \(\displaystyle \text{such that the sum of the distances of }P\text{ from }A\text{ and }B\text{ is a constant.}\)

\(\displaystyle \text{That is: }\A+ PB \:=\:2a\)

                  |     P
                  |   (x,y)
                  |     *
                  | *    *
                * |       *
       A    *     |        * B
    - - * - - - - + - - - - * - -
     (-5,0)       |       (5,0)
                  |

\(\displaystyle \text{The perimeter is }30,\;AB = 10.\)
. . \(\displaystyle \text{Hence:}\A + PB \:=\:20\quad\Rightarrow\quad 2a \:=\:20\quad\Rightarrow\quad a\,=\,10\)

\(\displaystyle \text{The "focal equation" is: }\:a^2\:=\:b^2+c^2\,\text{ and }c \,=\,5\)
. . \(\displaystyle \text{Hence: }\:10^2\:=\:b^2+5^2\quad\Rightarrow\quad b^2\:=\:75\)

\(\displaystyle \text{Therefore, the equation is: }\;\boxed{\frac{x^2}{100} + \frac{y^2}{75} \;=\;1}\)