Question 325483
<pre><b>
The standard forms for ellipses are

{{{(x-h)^2/a^2+(y-k)^2/b^2}}}{{{""=""}}}{{{1}}} for ellipses like this {{{drawing(50,25,-10,10,-5,5, arc(0,0,18,-8)   )}}} where a > b

{{{(x-h)^2/b^2+(y-k)^2/a^2}}}{{{""=""}}}{{{1}}} for ellipses like this {{{drawing(25,50,-5,5,-10,10, arc(0,0,8,-18)   )}}} where a < b

{{{25(x-2)^2 + 4(y+5)^2}}}{{{""=""}}}{{{100}}}

To get 1 on the right, divide through by {{{100}}}

{{{25(x-2)^2/100 + 4(y+5)^2/100}}}{{{""=""}}}{{{100/100}}}

Simplify:

{{{(x-2)^2/4 + (y+5)^2/25}}}{{{""=""}}}{{{1}}}

Since the larger number is under the expression in y the ellipse is 
of the form {{{(x-h)^2/b^2+(y-k)^2/a^2}}}{{{""=""}}}{{{1}}} 
and is like this {{{drawing(25,50,-5,5,-10,10, arc(0,0,8,-18)   )}}} 

{{{h=2}}}, {{{k=-5}}}, {{{a^2=25}}} or {{{a=5}}}, {{{b^2=4}}}, or {{{b=2}}}

center = (h,k) = (2,-5)

So let's begin by plotting the center (2,-5)

{{{drawing(1600/7,400,-2,6,-12,2, graph(1600/7,400,-2,6,-12,2),
line(2-.1,-5,2+.1,-5), line(2,-5-.1,2,-5+.1), locate(2,-5,"(2,-5)") 
  )}}}  

Next we draw a vertical line beginning at the center
(2,-5) and going upward {{{a=5}}} units, which is one-half the
major axis.  This ends in the point (2,0) which is the upper
vertex. 
 
{{{drawing(1600/7,400,-2,6,-12,2, graph(1600/7,400,-2,6,-12,2),
line(2-.1,-5,2+.1,-5), line(2,-5-.1,2,-5+.1), locate(2,-5,"(2,-5)"), green(line(2,-5,2,0)) 


 )}}}




Next we draw a vertical line beginning at the center
(2,-5) and going downward {{{a=5}}} units, which is one-half the
major axis.  This ends in the point (2,-10) which is the lower
vertex.  That green line is the major axis, and it is 10 units long.
 
 
{{{drawing(1600/7,400,-2,6,-12,2, graph(1600/7,400,-2,6,-12,2),
line(2-.1,-5,2+.1,-5), line(2,-5-.1,2,-5+.1), locate(2,-5,"(2,-5)"),
  green(line(2,0,2,-10)) 


 )}}}

Next we draw a horizontal line beginning at the center
(2,-5) and going to the right {{{b=2}}} units, which is one-half the
minor axis. This ends in the point (4,-5) which is the right co-vertex. 
 
{{{drawing(1600/7,400,-2,6,-12,2, graph(1600/7,400,-2,6,-12,2),
line(2-.1,-5,2+.1,-5), line(2,-5-.1,2,-5+.1), locate(2,-5,"(2,-5)"),
  green(line(2,0,2,-10), line(2,-5,4,-5)) 


 )}}}

Next we draw a horizontal line beginning at the center
(2,-5) and going to the leftt {{{b=2}}} units, which is one-half the
minor axis. This ends in the point (0,-5) which is the left co-vertex.
The horizontal green line is is the minor axis, and it is 4 units long.
 
{{{drawing(1600/7,400,-2,6,-12,2, graph(1600/7,400,-2,6,-12,2),
line(2-.1,-5,2+.1,-5), line(2,-5-.1,2,-5+.1), locate(2,-5,"(2,-5)"),
  green(line(2,0,2,-10), line(0,-5,4,-5)) 

 )}}}

Now we can sketch in the ellipse:

{{{drawing(1600/7,400,-2,6,-12,2, graph(1600/7,400,-2,6,-12,2),
line(2-.1,-5,2+.1,-5), line(2,-5-.1,2,-5+.1), locate(2,-5,"(2,-5)"),
 arc(2,-5,4,-10), green(line(2,0,2,-10), line(0,-5,4,-5)) 


 )}}}

Finally we find the foci.  They are the two points on the major axis
w2hich are c units from the center, where c is calculated by

{{{c^2=a^2-b^2}}}
{{{c^2=5^2-2^2}}}
{{{c^2=25-4}}}
{{{c^2=21}}}
{{{c=sqrt(21)}}}

So we add {{{sqrt(21)}}} to the y-coordinate of the center
to find the upper focus, which is the point
(2,{{{5+sqrt(21)}}}), which is about (2,9.6),
marked in red below.

{{{drawing(1600/7,400,-2,6,-12,2, graph(1600/7,400,-2,6,-12,2),
line(2-.1,-5,2+.1,-5), line(2,-5-.1,2,-5+.1), locate(2,-5,"(2,-5)"),
 arc(2,-5,4,-10), green(line(2,0,2,-10), line(0,-5,4,-5)), 

red(

line(2-.1,-5+sqrt(21),2+.1,-5+sqrt(21)),

line(2,-5+sqrt(21)-.1,2,-5+sqrt(21)+.1),

line(2-.1,-5+sqrt(21)+.1,2+.1,-5+sqrt(21)-.1),

line(2-.1,-5+sqrt(21)-.2,2+.1,-5+sqrt(21)+.1)



)


 

)}}}
 
And we subtract {{{sqrt(21)}}} from the y-coordinate of the center
to find the lower focus, which is the point
(2,{{{5-sqrt(21)}}}), which is about (2,0.4),
also marked in red below.

{{{drawing(1600/7,400,-2,6,-12,2, graph(1600/7,400,-2,6,-12,2),
line(2-.1,-5,2+.1,-5), line(2,-5-.1,2,-5+.1), locate(2,-5,"(2,-5)"),
 arc(2,-5,4,-10), green(line(2,0,2,-10), line(0,-5,4,-5)), 

red(

line(2-.1,-5+sqrt(21),2+.1,-5+sqrt(21)),

line(2,-5+sqrt(21)-.1,2,-5+sqrt(21)+.1),

line(2-.1,-5+sqrt(21)+.1,2+.1,-5+sqrt(21)-.1),

line(2-.1,-5+sqrt(21)-.2,2+.1,-5+sqrt(21)+.1),

line(2-.1,-5-sqrt(21),2+.1,-5-sqrt(21)),

line(2,-5-sqrt(21)-.1,2,-5-sqrt(21)+.1),

line(2-.1,-5-sqrt(21)+.1,2+.1,-5-sqrt(21)-.1),

line(2-.1,-5-sqrt(21)-.2,2+.1,-5-sqrt(21)+.1)

))}}}

Edwin</pre>