Question 83212
{{{3 * x ^ 2 + y ^ 2 - 6 * x + 4 * y - 4    =0}}} Start with the given conic




{{{3 * x ^ 2 + y ^ 2 - 6 * x + 4 * y = 4   }}}Add 4 to both sides




{{{( 3 * x ^ 2 - 6 * x ) + ( y ^ 2 + 4 * y ) = 4   }}} Group like terms





{{{3 ( x ^ 2 - 2 * x ) + 1 ( y ^ 2 + 4 * y ) = 4 }}} Factor a 3 out of the first parenthesis
and factor a 1 out of the second parenthesis.





Now we must complete the individual squares inside the parenthesis:




-----------------------------------------------------------------------


 For instance to complete {{{3 ( x ^ 2 - 2 * x ) }}}, take half of -2 and square it (ie {{{(-2/2)^2=(-1)^2=1}}} to get 1. Now add 1 inside the parenthesis like this: 

{{{3 ( x ^ 2 - 2 * x + 1 ) }}}

Since you really added {{{3*1}}} to the entire left side, we must add {{{3*1}}} to the right side also. Now lets complete  {{{1 ( y ^ 2 + 4 * y )}}}


In order to complete {{{1 ( y ^ 2 + 4 * y )}}}, take half of 4 and square it (ie {{{(4/2)^2=(2)^2=4}}} to get 4. Now add 4 inside the parenthesis like this: 

{{{1 ( y ^ 2 + 4 * y + 4 )}}}

Since you really added {{{1*4}}} to the entire left side, we must add {{{1*4}}} to the right side also.


----------------------------------------------------------------------- 


{{{3 ( x ^ 2 - 2 * x + 1 ) + 1 ( y ^ 2 + 4 * y + 4 ) = - 4 + 3 + 4 * 1   }}} Complete the individual squares by taking half of the 2nd coefficient and squaring it (remember to add to both sides).


{{{3 ( x ^ 2 - 2 * x + 1 ) + 1 ( y ^ 2 + 4 * y + 4 ) = 1 1 }}} Combine like terms on the right side





{{{3 ( x - 1 ) ^ 2 + 1 ( y + 2 ) ^ 2 = 1 1 }}} Factor the individual groups on the left side


{{{( 3 ( x - 1 ) ^ 2 + 1 ( y + 2 ) ^ 2 ) / 1 1 = 1 1 / 1 1 }}} Divide both sides by 11





{{{3 ( x - 1 ) ^ 2 / 1 1 + 1 ( y + 2 ) ^ 2 / 1 1 = 1 }}} Break up the fraction into standard form {{{(x-h)^2/a^2+(y-k)^2/b^2=1}}}


So the equation is an ellipse where the center is (1,-2) 

Here's the graph

{{{drawing (500,500,-10, 10, -10, 10,
   ellipse(1,-2,2*1.91485421551268,2*3.3166247903554),
   circle(1,-2,0.08),
   circle(1,-2,0.05),
   graph( 500, 500, -10, 10, -10, 10,0)
   )
}}} graph of {{{3 ( x - 1 ) ^ 2 / 1 1 + 1 ( y + 2 ) ^ 2 / 1 1 = 1 }}} with the center (1,-2)