Question 81934
{{{2 * x ^ 2 - y ^ 2 + 2 * x + 1 0 * y - 4 1    =0}}} Start with the given conic




{{{2 * x ^ 2 - y ^ 2 + 2 * x + 1 0 * y = 4 1   }}}Add 41 to both sides




{{{( 2 * x ^ 2 + 2 * x ) + ( - y ^ 2 + 1 0 * y ) = 4 1  }}} Group like terms





{{{2 ( x ^ 2 + x ) - 1 ( y ^ 2 - 1 0 * y ) = 4 1    }}} Factor a 2 out of the first parenthesis
and factor a -1 out of the second parenthesis.





Now we must complete the individual squares inside the parenthesis:




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 For instance to complete {{{2 ( x ^ 2 + x )  }}}, take half of 1 and square it (ie {{{(1/2)^2=(0.5)^2=0.25}}} to get 0.25. Now add 0.25 inside the parenthesis like this: 

{{{2 ( x ^ 2 + x + 0 . 2 5 )  }}}

Since you really added {{{2*0.25}}} to the entire left side, we must add {{{2*0.25}}} to the right side also. Now lets complete  {{{- 1 ( y ^ 2 - 1 0 * y ) }}}


In order to complete {{{- 1 ( y ^ 2 - 1 0 * y ) }}}, take half of -10 and square it (ie {{{(-10/2)^2=(-5)^2=25}}} to get 25. Now add 25 inside the parenthesis like this: 

{{{- 1 ( y ^ 2 - 1 0 * y + 2 5 ) }}}

Since you really added {{{-1*25}}} to the entire left side, we must add {{{-1*25}}} to the right side also.


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{{{2 ( x ^ 2 + x + 0 . 2 5 ) - 1 ( y ^ 2 - 1 0 * y + 2 5 ) = - 4 1 + 0 . 2 5 * 2 + 2 5 * - 1    }}} Complete the individual squares by taking half of the 2nd coefficient and squaring it (remember to add to both sides).


{{{2 ( x ^ 2 + x + 0 . 2 5 ) - 1 ( y ^ 2 - 1 0 * y + 2 5 ) = 1 6 . 5    }}} Combine like terms on the right side





{{{2 ( x + 0 . 5 ) ^ 2 - 1 ( y - 5 ) ^ 2 = 1 6 . 5  }}} Factor the individual groups on the left side


{{{( 2 ( x + 0 . 5 ) ^ 2 - 1 ( y - 5 ) ^ 2 ) / 1 6 . 5 = 1 6 . 5 / 1 6 . 5  }}} Divide both sides by 16.5





{{{1 ( x + 0 . 5 ) ^ 2 / 8 . 2 5 + 1 ( y - 5 ) ^ 2 / - 1 6 . 5 = 1 }}} Break up the fraction into standard form {{{(x-h)^2/a^2+(y-k)^2/b^2=1}}}

So the equation is a hyperbola where the center is (-0.5,5) 


From the graph, you can see the center (blue dot) 




Here's the graph of {{{1 ( x + 0 . 5 ) ^ 2 / 8 . 2 5 + 1 ( y - 5 ) ^ 2 / - 1 6 . 5 = 1 }}}

{{{drawing (500,500,-10, 10, -10, 10,
   graph( 500, 500, -10, 10, -10, 10,1.4142135623731*x-1.4142135623731*-0.5+5),
   graph( 500, 500, -10, 10, -10, 10,-1.4142135623731*x--1.4142135623731*-0.5+5),
   green(circle(2.37228132326901,5,0.15)),
   green(circle(-3.37228132326901,5,0.15)),
   blue(circle(-0.5,5,0.15)),
   graph( 500, 500, -10, 10, -10, 10,sqrt((16.5-2(x+0.5)^2)/-1)--5,-sqrt((16.5-2(x+0.5)^2)/-1)--5)
   )
}}}  note: the straight lines are not part of the graph. They are the asymptotes with slopes of ±{{{b/a}}}.


Here is a check/verification graph of {{{2 * x ^ 2 - y ^ 2 + 2 * x + 1 0 * y - 4 1    =0}}}


{{{ graph( 500, 500, -10, 10, -10, 10, sqrt((16.5-2(x+0.5)^2)/-1)--5,-sqrt((16.5-2(x+0.5)^2)/-1)--5) }}}