## Identifying a Conic 2

To decide whether a general conic is an ellipse, parabola or hyperbola, we only need to completer the square. The form of the completed square will determine which type of conic it is.
Example: Which type is the conic
$x^2+4xy+y^2+2y=5$
,
Completing the square for the
$x$
coordinate gives the equation
u
$(x+2y)^2-(2y)^2+y^2+2y=5$

$(x+2y)^2-3y^2+2y=5$

Completing the square for the
$y$
coordinate gives the equation
$(x+2y)^2-3(y+1/3^2-3(1/3)^2=5$

Sim[ifying the equation gives
$(x+2y)^2-(\frac{y-1/3}{1/ \sqrt{3}})^2=16/3/2$

This is the form of a hyperbola. The axes of the ellipse are
$x+2y=0$
and
$\frac{y-1/3}{\sqrt{3}}=0$
.