SOLUTION: The problem is: Sketch a graph of this equation, find the coordinates of the foci, and find the lengths of the major and minor axes. 3x^2+2y^2=24 I know it may be hard to

The problem is: 
Sketch a graph of this equation, find the coordinates of the foci, 
and find the lengths of the major and minor axes. 
3x^2+2y^2=24 
I know it may be hard to show how to graph this, so just putting 
it into the correct form of an ellipse, and then helping me figure 
out what the foci and axes are would be very helpful! Thanks :)!!

We use the fact that the equations of ellipses with centers
at the origin are of two different forms:

 for ellipses which look like this 

,

and have foci (±,0) where 

and

 for ellipses which look like this 



and have foci (0,±) where 

You can tell which is which by looking at
the denominators. It is always true that
 is larger than .

 represents the semi-major axis, or as some say,
the "long radius"

 represents the semi-minor axis, or as some say,
the "short radius"

--------

Starting with your equation,



we must get it into one of the forms above.  So
we must first get a 1 on the right by dividing
every term through by 24:



which simplifies to



Since 12 is larger than 8, we know that
 and  and so the
ellipse is of the form

 and looks something
like the second figure above.

Since it has foci (0,±) where ,
then
,
, 
,


The foci are 
(,±)

Now we'll draw the graph, since 
Then  = ,
or about  so we draw the major axis
from the vertex point (,) to the
other vertex point (,)

    

Then we draw the minor axis from (-b,0) to (b,0)
and since , then , or
, or , or
about  
 so we
draw the minor axis from (,) to
(,)

 

Now we can sketch in the ellipse and also
indicate the two foci,  
(, ±)
with two short lines

 

Edwin