SOLUTION: i'm learning about ellipses and i have the problem: x squared+4y squared-6x+24y+41=0 i need to find the center, foci, and vertices. i have no idea how to even start, i tried by

Get the number on the right by adding  to both sides:



Swap the two middle terms to get the -term next to the -term


Multiply the coefficient of x, which is , by ,
get  then square that number  which 
gives 

Now add  to both sides:



Since the  term has a coefficient other than 1,
factor the 4 out of the last two terms on the left:



Now we complete the square inside the parentheses.

Multiply the coefficient of y, which is , by ,
get  then square that number  which 
gives 

Now we add  inside the parentheses.  However
when we add  inside the parentheses we are REALLY
adding  or  to the left side 
because everything in the parentheses is multiplied by the 
in front of the parentheses.  So we add  to the right side:



Now we factor the first three terms as  or 
and we factor the three terms inside the parentheses as 
or , and combine the numbers on the right as 



Now to get 1 on the right we divive every term through by 4





Notice that I left a 1 under the bottom of the second term
so that it would be in standard form

Since  is larger than  and the larger term is under
the  term it is in the form

 

and has its major axis horizontal.

, , , so 

, so , so we plot the center {3,-3)



Then we draw the major axis by drawing a horizontal line 
units both right and left of the center (3,-3)



Now we can get the vertices, for they are the points ar the ends
of the major axis, and we can see that the major axis, the green
line segment starts at (1,-3) on the left and goes to (5,-3) on
the right, so the vertices are

(1,-3) and (5,-3)

Now we draw the minor axis by drawing a horizontal line 
units both up and down from the center (3,-3):



Now we can sketch in the ellipse:



Finally we need to find the foci.  The distance from
the center to each focus is  and  is
calculated by this Pythagorean identity:









The coordinates of the foci are
points which are distance of 
units right and left of the center on the major axis,

so the two foci are these points: 

(3-Ö3,-3) and (3+Ö3,-3)

They are plotted below:



Edwin