SOLUTION: how can i find out the equation for ellipse when i have foci and length of major axis is given?

Question 1045317: how can i find out the equation for ellipse when i have foci and length of major axis is given?
Found 2 solutions by KMST, ikleyn:
Answer by KMST(5328)   (Show Source): You can put this solution on YOUR website!
You also need to know the coordinates of the center, and
what direction the major axis follows.

An ellipse is a stretched circle.
A circle of radius with center at the origin has the equation
<---> ,
because is the square of the distance from point (x,y) to origin (0,0) .
Thankfully, we almost always get confronted by ellipses that are easier to calculate,
where the longest axis of symmetry, the major axis, is either
parallel to the x-axis (and we call it horizontal), or
parallel to the y-axis (and we call it vertical).
An ellipse with center at the origin and axes of symmetry parallel to the x- and y-axes has the equation
or , with .
If the center of a circle, or ellipse is not ,
but a point ,
you just write instead of ,and
instead of .

The distance from each focus to the center, is related to and by , so you only need to know two of those distances.
Given foci and length of major axis, you have and ,
just find and plug the found value into the equation.

.

The points the farthest from the center are the vertices,
located on the major axis, at a distance from the center.
The points the closest to the center are often called co-vertices,
located on the minor axis, at a distance from the center.
The foci are located on the major axis, at a distance to either side of the center.

Here is an ellipse. you see the origin, point .
You see one labeled vertex, point from the center.
I labeled the foci, focus , and Focus ,
both at a distance from the center.
I also labeled one of the co-vertices, point , at a distance from the center.
The fancy definition of ellipse says that for all the points on the ellipse,
the sum of the distances to one Focus and the other is the same.

You can see that point is at a distance from focus ,
and at a distance ( from focus .
So the sum of the distance to the foci is
for point , and for all points in the ellipse.
Then, point is at the same distance from focus and focus .
Applying the Pythagorean theorem to right triangle ,
you get the relationship .
Answer by ikleyn(52803)   (Show Source): You can put this solution on YOUR website!
.
On introductory knowledge for an ellipse see also the lesson
    - Ellipse definition, canonical equation, characteristic points and elements
in this site.

Also see the lesson
    - Find a standard equation of an ellipse given by its elements.

Although it does not contain exactly what you need, it may be of help to you.

And the last notice: If you want to get more specific answer, send us your specific data.

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