there may be more efficient way but one can go step by step and construct points, then solve ellipse using simultaneous equations:
1. find center (interpolation):
x=[(-3)+(5)]/2=1
y=[(-2)+(-3)]/2=-2.5
center is (1,-5/2)
2. find line that goes through both foci (just a warm up exercise):
y=mx+b // line representation
m= slope
b = y intercept
m=(y2-y1)/(x2-x1)= -1/8
using known point on the line (one of foci such as x=5, y=-3) we can determine b:
y=mx+b
-3=(-1/8)*5+b
b= -19/8
so line is
y= -(x+19)/8
3. find point(s) on major axis. we know that this is along line that passes through both foci and distance from centre is twice the distance of foci from centre
suppose we look at foci point (5,-3) and we know centre is at (1, -5/2). then distance between them (along x and y axes) is
dx=5-1=4
dy=-3-(-5/2)=-1/2
therefore, point on major axis close to that foci is
(5,-3)+(4, -1/2) = (9, -7/2)
to verify result, we can check if this is on the line we just calculated
-7/2=-(9+19)/8
-7/2=-28/8
which is true
this is one of the points on ellipse, mark it for later use: (9, -7/2)
4. do the same for other foci, note dx and dy are same but displacement is now in opposite direction from the foci (hence negative sign):
(-3,-2) - (4, -1/2)= (-7, -3/2) // this should be the second point, verify it
5. find line that is normal to one going through foci (you had warmup, now do this on your own). we know that slope of perpendicular line is
m2 = -1/m1 = -1/(-1/8) = 8
and we know that this line also must go through center of the ellipse (so we have reference point and can calculate intercept)
6. find point that is on this line (minor axis line as found in step 5) and that is same distance from center as one of the foci.
the other way to deal with this is to just swap dx and dy:
dx=-1/2
dy=4
and add or subtract them from center (1,-5/2)
(1,-5/2) + (-1/2,4) = (1/2, 3/2) // this is another point on ellipse
(1, -5/2) - (-1/2, 4) = (3/2, -13/2) // this is another point on ellipse
7. so far we have found several points on ellipse. now you could go to wiki and read about ellipse and all the goodies like eccentricity etc or you could just write formula and solve it for known points (system of equations:
x^2/a^2 - y^2/b^2=1