When the major axis is horizontal, the standard equation of an ellipse is
#(x - h)^2/a^2 + (y - k)^2/b^2 = 1#
where
#C: (h, k)#
#V: (h +- a, k)#
#c^2 = a^2 - b^2#
#f: (h +- c, k) #
On the other hand, when the major axis is vertical, the standard equation of an ellipse is
#(x - h)^2/b^2 + (y - k)^2/a^2 = 1#
where
#C: (h, k)#
#V: (h, k +- a)#
#c^2 = a^2 - b^2#
#f: (h, k +- c)#
In the given,
#C: (-3, 1)#
#V_1: (-3, 3)#
#f_1: (-3, 0)#
Since the #x#-coordinate of the points are constant, we can say that the major axis is vertical.
#C: (h, k)#
#C: (-3, 1)#
#=> h = -3#
#=> k = 1#
#V: (h, k +- a)#
#V_1: (-3, 3)#
#=> 3 = k +- a#
#=> 3 = 1 +- a#
#=> 3 = 1 + a#
#=> 3 = 1 - a#
#=> 2 = a#
#=> -2 = a#
Since #a# is the distance between the center and the vertex, we take #a# to be positive
#=> a = 2#
#f: (h, k +- c)#
#=> 0 = 1 +- c#
#=> c = 1#
#=> c = -1#
Similarly for #c#, since it is the distance between the focus and the center, we take #c# to be positive
#=> c = 1#
#c^2 = a^2 - b^2#
# 1^2 = 2^2 - b^2#
#=> 1 = 4 - b^2#
#=> b^2 = 3#
#=> b = +-sqrt3#
Again, since #b# is the distance between the center and the end of the minor axis (not sure what it was called), we take #b# to be positive
#=> b = sqrt3#
Hence, the equation of the ellipse is
#(x - -3)^2/2^2 + (y - 1)^2/(sqrt3)^2 = 1#
#=> (x + 3)^2/4 + (y - 1)^2/3 = 1#
