Question 1166898
<pre>
Here's one exactly like your third problem.</pre>C(0,0), a vertex is 18 units from a focus and 32 units from the other.<pre>The foci are (-c,0) and (c,0) and the right vertex is at (a,0)

The distance from (c,0) to (a,0) is a-c=18
The distance from (-c,0) to (a,0) is a-(-c)=32

So we solve the system of equations:

{{{system(a-c=18,a-(-c)=32)}}}

and get a=25, and c=7

So the foci are at (-7,0) and (7,0) and the vertices are (25,0) and (-25,0) 

So we have this graph

So the graph is below.  It looks more like a circle because the foci are closer
together than usual.  But it is a little wider than it is tall.

{{{drawing(800,800,-30,30,-30,30, graph(800,800,-30,30,-30,30),
arc(0,0,50,-48), circle(-7,0,.5),circle(7,0,.5),circle(25,0,.5),circle(-25,0,.5) )}}}

It has the equation:

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

The center is (h,k) = (0,0), and a = 25

So we have this much of the equation:

{{{(x-0)^2/25^2+(y-0)^2/b^2=1}}}

{{{x^2/625^""+y^2/b^2=1}}}

We find b by the Pythagorean relation for all ellipses:

{{{c^2=a^2-b^2}}}
{{{7^2=25^2-b^2}}}
{{{49=625-b^2}}}
{{{b^2=576}}}
{{{b=24}}}

So the equation is

{{{x^2/625^""+y^2/576^""=1}}}

Now do yours step-by-step the same way.

Edwin</pre>