Locate the foci with this equation:


(x - 5)²    (y - 3)²
———————— + ————————— = 1
   5²         10²

There are two forms of ellipses.

-------------------------------------------------------------
1. Those that are longer horizontally and narrower vertically

These have the form:

(x - h)²   (y - k)²
———————— + ———————— = 1
   a²         b²

a = semi-major axis, b = semi-minor axis, (h, k) = center

center = (h, k)
vertices = (h±a, k)        _______
foci = (h±c, k) where c = Öa² - b² 

------------------------------------------------------------

2. Those that are longer vertically and narrower horizontally

(x - h)²   (y - k)²
———————— + ———————— = 1
   b²         a²

a = semi-major axis, b = semi-minor axis, (h, k) = center

center = (h, k)
vertices = (h, k±a)        _______
foci = (h, k±c) where c = Öa² - b² 

--------------------------------------------------------------

You can always tell which type ellipse you have because
the semi major axis a is always greater than the semi minor
axis b.  Therefore a² will always be larger than b².  If
a² is under the (x-h)², the ellipse is the first type. Otherwise
it is the second type.

Yours is the second type because the larger of 5² and 10² is
10² and it is underneath (y-k)².

a = 10
b = 5
center = (h, k) = (5. 3)
vertices = (h, k±a) = (5, 3±10), that is, (5, -7) and (5, 13) 
    
To find the foci, we need to find c
     _______
c = Öa² - b² 
     ________
c = Ö10² - 5² 
     ________
c = Ö100 - 25
     __
c = Ö75
     ____
c = Ö25·3
      _
c = 5Ö3
                          _
foci = (h, k±c) = (5, 3±5Ö3), that is
        _              _
(5, 3-5Ö3) and (5, 3+5Ö3)

Your ellipse looks like this:



Edwin
AnlytcPhil@aol.com