## Identifying a Conic

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
$x^2+4x+2y^2+6y=5$
,
Completing the square for the
$x$
coordinate gives the equation
$(x+2)^2-(2)^2+2y^2+6y=5$

Completing the square for the
$y$
coordinate gives the equation
$(x+2)^2-(2)^2_2+2(y+3/2)^2-2(3/2)^2=5$

Sim[ifying the equation gives
$(x+2)^2+2(y+3/2)^2=(x+2)^2+(\frac{y+3/2}{\sqrt{2}})^2=27/2$

This is the form of an ellipse. The axes of the ellipse are
$x+2=0$
and
$\frac{y+3/2}{\sqrt{2}}=0$
, and the major axis is
$\frac{y+3/2}{\sqrt{2}}=9$
.