SOLUTION: I am having trouble with writing an equation of an ellipse, center is at the origin. Focus at (2,0); 7/3 is one-half length of the minor axis.

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Question 1103542: I am having trouble with writing an equation of an ellipse, center is at the origin. Focus at (2,0); 7/3 is one-half length of the minor axis.
Answer by Edwin McCravy(20065) About Me  (Show Source):
You can put this solution on YOUR website!
We plot the center and focus:



So we know that c = 2 since c is the distance 
from the center to the focus.  

We also know that the ellipse is like this drawing%2820%2C10%2C-2%2C2%2C-1%2C1%2Carc%280%2C0%2C-3.9%2C1.9%29+%29
so its equation is x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2=1

We also know that the other focus is 2 units on the 
left of the center, at (-2,0) so we plot that too:



Since we know that 7/3 is one-half length of the minor axis,
that means that that b = 7/3, the semi-minor axis.

So we know that the top point (sometimes called the top covertex)
is (0,7/3), so we plot that too, as well as the other covertex at
(0,-7/3):



The ellipse equation for a,b, and c is a%5E2-b%5E2=c%5E2, so

a%5E2-b%5E2=c%5E2
a%5E2-%287%2F3%29%5E2=%282%29%5E2
a%5E2=%282%29%5E2%2B%287%2F3%29%5E2
a%5E2=4%2B49%2F9
a%5E2=36%2F9%2B49%2F9
a%5E2=85%2F9
a=sqrt%2885%29%2F3

That's about 3.073, so we plot the vertices, too,
at %22%22+%2B-+sqrt%2885%29%2F3



So the equation 

x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2=1

becomes:

x%5E2%2F%2885%2F9%29%2By%5E2%2F2%5E2=1

x%5E2%2F%2885%2F9%29%2By%5E2%2F4=1

You can simplify it to 

36x%5E2+%2B+85y%5E2+=+340

if you like.



Edwin