Question 310657
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There are two types of equations for ellipses

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

where the ellipse looks like the cross section of an egg
resting on a table.

and

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

where the ellipse is upright like the number zero "0".

In either case a is half the major axis and b is half the minor axis.

The major axis is always larger than the minor axis.  So a > b.

In the case where  a = b, the ellipse is a circle.

The center is the point (h,k).  The foci are two points 

inside the ellipse on the major axis which are c units from the 

center, where c is gotten from the equation {{{c^2=a^2-b^2}}}.

Your ellipse

{{{(x+5)^2/9 +(y-7)^2/25=1}}}

is the type that is upright like the letter zero "0", because the
larger denominator is under the term in y.  So we compare it to

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

and we see that h=-5, k=7, {{{a^2=25}}} so {{{a=5}}} and {{{b=3}}}

So the center is (-5,7), the major axis is 2*5 or 10 and the minor
axis is 2*7=14, so we draw the graph:

{{{drawing(300,400, -10,2,-2,14, 

graph(300,400, -10,2,-2,14), arc(-5,7,6,-10) )}}}

We draw in the major and minor axes, which cross at the center:

{{{drawing(300,400, -10,2,-2,14, green(line(-5,2,-5,12)),
green(line(-8,7,-2,7)),locate(-5,7,"C(-5,7)"),
graph(300,400, -10,2,-2,14), arc(-5,7,6,-10) )}}}

The foci are on the major axis and are c units above and
below the center.  We calculate c

{{{c^2=a^2-b^2}}}
{{{c^2=5^2-3^2}}}
{{{c^2=25-9}}}
{{{c^2=16}}}
{{{c=4}}}

So the foci are 4 units directly above and below the center (-5,7).

One focus is at (-5,11) and the other is at (-5,3)


{{{drawing(300,400, -10,2,-2,14, green(line(-5,2,-5,12)),
green(line(-8,7,-2,7)),locate(-5,7,"C(-5,7)"),
line(-5-.1,11,-5+.1,11), line(-5-.1,3,-5+.1,3), locate(-5,11,"F(-5,11)"),
locate(-5,3,"F(-5,3)"),

graph(300,400, -10,2,-2,14), arc(-5,7,6,-10) )}}}

You didn't ask for the vertices and the co-vertices.

They are at the ends of the major and minor axes.

The vertices are (-5,2) and (-5,12)
The covertices are at (-8,7) and (-2,7)

Edwin</pre>