Question 292355
{{{2x^2+3y^2+4x-12y-4=0}}}
<pre><font size = 4 color = "indigo"><b>
Rearrange

{{{2x^2+4x+3y^2-12y=4}}}

Factor out coefficients of squared letters:

{{{2(x^2+2x)+3(y^2-4y)=4}}}

Complete the square in the first parentheses by
adding {{{red("+1")}}} inside the first parentheses
which actually amounts to adding 2 to the left side 
because there is a 2 in front of the parentheses, so
we must add a 2 to the right side:

{{{2(x^2+2x+red(1))+3(y^2-4y)=4+red(2)}}}

Complete the square in the second parentheses by
adding {{{red("+4")}}} inside the second parentheses
which actually amounts to adding 12 to the left side 
because there is a 3 in front of the parentheses, so
we must add a 12 to the right side:

{{{2(x^2+2x+red(1))+3(y^2-4y+red(4))=4+red(2)+red(12)}}}

Factoring the parentheses as perfect squares:

{{{2(x+1)^2+3(y-2)=18}}}

Get a 1 on the right by dividing through by 18

{{{(2(x+1)^2)/18+(3(y-2)^2)/18=18/18}}}

{{{(x+1)^2/9+(y-2)^2/6=1}}}

Since the largest denominator is under the term in
x, the ellipse has a horizontal major axis.  So we
compare it to:

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

{{{h=-1}}}, {{{k=2}}}, 

{{{a^2=9}}} so {{{a=sqrt(9)=3}}}

{{{b^2=6}}} so {{{b-sqrt(6)}}}

Its center is at (h,k) = (-1,2)

Plot the center:

{{{drawing(400,400,-7,6,-7,6,
graph(400,400,-7,6,-7,6),
line(-1+.1,2,-1-.1,2), line(-1,2+.1,-1,2-.1),
line(-1+.1,2+.1,-1-.1,2-.1), line(-1+.1,2-.1,-1-.1,2+.1)

  )}}}
 
Draw the major axis {{{a=3}}} units both right and left of the center.
Draw the minor axis {{{b=sqrt(6)=2.449489743}}} units up and down
from the center.


Connect them to show the major and minor axes
of the ellipse:
 
{{{drawing(400,400,-7,6,-7,6,
graph(400,400,-7,6,-7,6), line(-4,2,2,2), line(-1,2-sqrt(6),-1,2+sqrt(6)) )}}}
 
Sketch in the ellipse:
 
{{{drawing(400,400,-7,6,-7,6,
graph(400,400,-7,6,-7,6), line(-4,2,2,2), line(-1,2-sqrt(6),-1,2+sqrt(6)), 
arc(-1,2,6,-2sqrt(6))

)}}}

 
Edwin</pre>