We have talked about writing equations of ellipses with the center at (0,0). Do you remember what two variables you need to find to write an equation of an ellipse? Check below to see if you are correct.

If you said a² and b² you are correct. Well, when the center is not (0,0) you still have to find a² and b², but now you also need to find the center. We do this by applying the midpoint formula. Let's explore the situation where we are given the two foci of (-6,1) and (-2,1) and the sum of the focal radii is 6. We must still use the distance between the two foci to find 2c and 2a is equal to the sum of the focal radii. So can you tell me what you would get for a and c, and thus a² and c²? Check below and see if you found the correct answers so far.

2a = 6, so a =3. 2c = 4, so c = 2. ( You subtract the two coordinates that are different to find the distance between the foci since they are on a horizontal line.) This makes a² = 9 and c²=4. Also, since the foci are on a horizontal line....the major axis is the x axis so if you recall from before, the major axis will then be the x axis. This means that a² should go under the x variable and b² should go under the y variable in the equation of an ellipse just like before. But we must find b² first by using the formula b² = a² - c²....giving us b² = 9 - 4 which is b² = 5.

However we must also find the center and we do this by using the midpoint formula. Do you recall what the midpoint formula is? Check below to see if you are correct.

The midpoint formula is:

( x₁+x₂ y₁+y₂ )
------ , -------
2 2

so if we add -6 + -2 from the x coordinates of the foci and divide by 2 and then add the 1 + 1 from the y coordinates and divide by 2, we get the center of the ellipse is (-4, 1). When we write the equation of the ellipse we subtract the x and y coordinate from the x and y variable. So can you write down what the equation will look like since we now have the center, a² and b²? Check down below and see if your final answer is correct.

Putting all the information together....our equation should be:

(x- - 4)²      (y - 1)²
----------- + ----------- = 1
    9              5

and simplifying one last thing our very final equation should be:

(x + 4)²     (y -1)²
---------- + ----------- = 1
    9             5