The first case we will consider is when we are given the x intercepts and the y intercepts. This is the easiest to work with because all you have to do is square both intercepts and put them below the appropriate variables.

For example, lets take the situation where the x-intercepts are 4 and -4. (Remember an intercept is where the ellipse crosses the x axis). Think in reverse from when we graphed ellipses. We took the square root of the number below each variable and these were the points we plotted on the x and y axis to obtain our ellipse....so now we do the opposite and square them to put them into the equation. And if the y-intercepts are 7 and -7, we do the same.

Can you guess how that might look? Try it and check your answer below.

We square 4 and -4 and get 16. Then we square 7 and -7 and get 49. We place these in the equation like this:

x²      y²
---- + ----- = 1
16     49

Is this your answer...then you are correct.

Now lets take another situation where the foci and "the sum of the focal radii" is given. Remember the sum of the focal radii represents the major axis and this is always 2a. The distance between the foci is 2c. So if the foci are (0,2) and (0,-2) and the sum of the focal radii is given to be 12, can you find a and c? Continue to see if you are correct.

2c would equal 4 (the distance from 2 to -2) and 2a would equal 12, making c = 2 and a = 6. Were you correct?

Now square these two numbers and place them in the appropiate places in the equation: b² = a² - c² and solve for b². What did you get for b²? Check below.

Substituting we should get this: b² = a² - c²
b² = 36 - 4
b² = 32

Now there is just a little work left to do. Do you remember that the major axis is the axis that the foci are on? So can you tell where you should place the numbers to obtain the correct equation? Try it and then check your answer below.

Since the foci are on the y axis, the y axis is the major axis. Therefore the a² should be placed under the y variable and the b² should be placed under the x variable like this:

x²      y²
---- + ---- = 1
32     36

And this is your final answer.