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  
\[\frac{y-1/3}{\sqrt{3}}=0\]
.

Add comment

Security code
Refresh