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## Homework Statement

Hello, my friend asked my If I could help him with this problem. However I just can't seem to find a way to solve this.

Ellipse

Focus(2,2)

vertex(2,-6)

Point(26/5,2)

a+e=8

find the equation of the ellipse

## Homework Equations

(x-m)^2/a^2+(y-n)^2/b^2=1

Center(m,n)

a=moyor axis

b=minor axis

## The Attempt at a Solution

I will post pictures of my work as I don't yet know how to use math syntax on the internet.

I would really appreciate if you could find my mistake. I tried duing everything, however I can't seem to get the point(26/5,2) to be on my ellipse. Everything else looks fine

The order of which I did the problem is represented by roman numbers

So what I tried doing was:

From the definition I know that the sum of the distance from each of the two fucoses to the outer line is constant and it equals 2a. We are also told that a+e=8. From the picture we can clearly see that the distance from the upper tocus to the vertex equals a+e( or 8). And we also know that 2a=8+e solving both equation we get that a=16/3 and e=8/3. Then I used the pitagoras therom to get b( the minor axis) and from then on I used the point (2,-6) in the ellipse equation to get the center( later I realized that I could just substract e from the first focus). I got the final equation, however when I go to check it with the point (26/5,2) I don't get the right answer.

Below I will include the picture of the ellipse I made using the desmos graphing calculator

as you can see the point does not lay on the line.

I would really appreciate it if you could check my work and find my mistake.