How do I find the equation of an ellipse with:

Vertices: (-7,10), (-7,-10)
Foci: (-7, square root of 19) (-7, - square root of 19)

I think the center is (-7,0) and a^2 is 49 I just don't know how to get c^2 so I cant get b^2

Looks like you label your a to be associated with the major axis (the axis containing the foci)

In that case:
b^2 + c^2 = a^2

b^2 + (√19)^2 = 7^2
b^2 + 19 = 49
b^2 = 30

so are correct about the centre, so

(x+7)^2 /30 + y^2/49 = 1

http://www.mathwarehouse.com/ellipse/equation-of-ellipse.php

To find the equation of an ellipse, you need to know the coordinates of the center, the lengths of the major and minor axes (or the values of a and b), and the orientation of the ellipse. In this case, you already correctly determined that the center is (-7, 0) and that a^2 is 49.

To find c^2, which is the square of the distance from the center to each focus, you can use the distance formula. The formula is given by:

c^2 = a^2 - b^2

where a is the length of the semi-major axis, b is the length of the semi-minor axis, and c is the distance from the center to each focus.

In this case, a^2 is given as 49. To find b^2, you can use the relationship between c^2, a^2, and b^2:

c^2 = a^2 - b^2

Since you already know c^2 and a^2, you can rearrange the equation and solve for b^2:

b^2 = a^2 - c^2

Plugging in the values, you have:

b^2 = 49 - c^2

Now, let's find c^2. The foci are given as (-7, sqrt(19)) and (-7, -sqrt(19)). The x-coordinate remains the same since the center is (-7, 0). Thus, the distance between the center and each focus is the absolute value of the difference between the y-coordinates:

c = |sqrt(19) - 0| = sqrt(19)

Now, you can substitute c^2 = 19 into the equation for b^2:

b^2 = 49 - 19
b^2 = 30

Now that you have a^2 = 49 and b^2 = 30, you can write the equation of the ellipse in standard form:

(x - h)^2/a^2 + (y - k)^2/b^2 = 1

Substituting the values of the center (-7, 0) and a^2 = 49, b^2 = 30, you get:

(x + 7)^2/49 + y^2/30 = 1

So, the equation of the ellipse is (x + 7)^2/49 + y^2/30 = 1.