The equation of the ellipse having foci at(-2,-3)and (-2,5)and a major axis of length 10 is?

The center of the ellipse is the midpoint of the foci, which is (-2, 1). The distance between the foci is 8 (5-(-3)), which is also the length of the major axis.

The equation of an ellipse with center at (h, k), major axis of length 2a, and foci at (h+c, k) and (h-c, k) is:

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

In this case, h = -2, k = 1, a = 4 (half of the major axis length), and c = 4 (half of the distance between the foci).

Plugging in these values, the equation of the ellipse is:

(x+2)^2/16 + (y-1)^2/9 = 1

The coordinates of the foci are (-2,-3) and (-2,5).
The distance between the foci is 2c = 10, so c = 5.
The length of the major axis is 2a = 10, so a = 5.
The equation of the ellipse is:

$$\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$$

where (h,k) are the coordinates of the center of the ellipse.

The center of the ellipse is the midpoint of the foci, so:

$$(h,k) = \left(\frac{-2+(-2)}{2}, \frac{-3+5}{2}\right) = (-2,1)$$

Substituting the values of a, b, h, and k into the equation of the ellipse, we get:

$$\frac{(x+2)^2}{25}+\frac{(y-1)^2}{16}=1$$