Question 117291
<pre><font size = 3><b>
Graph the ellipse and find the coordinates of the center 
vertices and foci.

{{{25x^2+16y^2}}} = 400

equations of the form {{{Mx^2+Ny^2=P}}} have their center at the
origin.  So the center is (0,0)

{{{25x^2+16y^2}}} = 400

We want to get this either to the form

{{{x^2/a^2 + y^2/b^2}}} = 1 or {{{x^2/b^2 + y^2/a^2}}} = 1

The first one is shaped like an egg sitting on a table.
The second one has the shape of the number zero, like this " 0 ".
We will know which form it is in because {{{a^2}}} is always larger
than {{{b^2}}}.

{{{25x^2+16y^2}}} = 400

Get 1 on the right by dividing through by 400:

{{{(25x^2)/400+(16y^2)/400}}} = {{{400/400}}}

{{{x^2/16+y^2/25}}} = 1

The larger denominator on the left side is 25,
so a² = 25, the smaller denominator of the left
is 16, so b² = 16.

So this graph is in the form {{{x^2/b^2 + y^2/a^2}}} = 1

and it will have the shape of a 0.

Since a² = 25, a = 5, Since b² = 16, b = 4

The center is at the origin.

One half the major axis extends from (0,0) to (0,5),
and the other half extends from (0,0) to (0,-5).

One half the minor axis extends from (0,0) to (4,0),
and the other half extends from (0,0) to (-4,0).

So we draw an upright rectangle through those four 
points, like this:

{{{drawing(200,200,-6,6,-6,6, graph(200,200,-6,6,-6,6), 
rectangle(-4,-5,4,5) )}}} 

Draw an upright ellipse just fitting in that rectangle,
shaped like a zero "0":

{{{drawing(200,200,-6,6,-6,6, graph(200,200,-6,6,-6,6,sqrt(400-25x^2)/4), 
 graph(200,200,-6,6,-6,6,-sqrt(400-25x^2)/4  ), rectangle(-4,-5,4,5) )}}}

It's vetices are the "bluntest" points on the ellipse.
They are (0,5) and (0,-5)

Erase the rectangle:

{{{drawing(200,200,-6,6,-6,6, graph(200,200,-6,6,-6,6,sqrt(400-25x^2)/4), 
 graph(200,200,-6,6,-6,6,-sqrt(400-25x^2)/4  ) )}}}

Now we calculate the value of <font color = "red">c</font> which in the distance
from the <font color = "red">c</font>enter to the fo<font color = "red">c</font>i.  You can remember what
<font color = "red">c</font> is by noticing that the words "fo<font color = "red">c</font>us", "fo<font color = "red">c</font>i", and
"<font color = "red">c</font>enter" contain the letter "<font color = "red">c</font>".

The formula is <font color = "red">c</font>² = a² - b²
               <font color = "red">c</font>² = 25 - 16
               <font color = "red">c</font>² = 9
                <font color = "red">c</font> = {{{sqrt(9)}}}
                <font color = "red">c</font> = 3

So the fo<font color = "red">c</font>i are at (0,3) and (0,-3) marked below with
short lines at those points:

{{{drawing(200,200,-6,6,-6,6, graph(200,200,-6,6,-6,6,sqrt(400-25x^2)/4), 
 graph(200,200,-6,6,-6,6,-sqrt(400-25x^2)/4),line(-.6,3,.6,3), line(-.6,-3,.6,-3) )}}}   

Edwin</pre></font></b>