When you only have the foci and the length of the major axis, how do you find the equation of an ellipse?

Question

The foci are (-2,1) and (-2,-3), and the length of the major axis is 18. Please help!!!

Answer 1

The general equation of an ellipse is: (x-h) ² / a ² + (y-k)² / b² = 1 Where (h,k) is the center. You can find the center, it is half way between the foci. The center is (-2,-1) We can plug that in the equation at this point and simplify

(x-(-2)) ² / b ² + (y-(-1))² / a² = 1

(x+2) ² / b ² + (y+1)² / a² = 1

We also know the length of the major axis, with that we can calculate a.

a=18/2

a=9

a²= 9² = 81

(x+2) ² / b ² + (y+1)² / 81 = 1

The last thing we need is to find b². We can use the foci equation to do this. c² = a² - b² or b² = a² - c² if we rearrange the equation.

We know that the distance between the foci is 4.

c= 4/2

c = 2

b² = a² - c²

b² = 9² - 2²

b² = 81-4 = 77

Last step is to plug that value into the equation.

(x+2) ² / 77 + (y+1)² / 81 = 1