0

My math teacher told me today that the graph of the equation $ax^2+2hxy+by^2+2gx+2fy+c=0$ can represent more than just a pair of straight lines, as it can be a pair of straight lines only if the determinant:

$\begin{vmatrix}a & h & g\\h & b & f\\g & f & c\end{vmatrix}=0$

Can it also represent graphs other than those of straight lines?

Abhigyan
  • 477
  • 5
  • 19
  • 4
    The magic word to look at is conics – Francesco Polizzi Aug 24 '17 at 14:21
  • How about $a=b=1$, $c=-1$, $f=g=h=0$? – Angina Seng Aug 24 '17 at 14:23
  • 2
    It's the general equation of a conic. This conic is degenerate if the determinant is $0$. In this case you obtain either a pair of straight lines, or a double straight line, depending on the rank of the corresponding matrix. – Bernard Aug 24 '17 at 14:23
  • @Bernard Is it possible to list out all of the cases? If yes, then what are the specific cases of a parabola, hyperbola, ellipse, circle, etc.? – Abhigyan Aug 24 '17 at 14:46

1 Answers1

1

HINT.- You do have $$ax^2+2hxy+by^2+2gx+2fy+c=\begin{bmatrix}x&y&1\end{bmatrix}\begin{bmatrix}a&h&g\\h&b&f\\g&f&c\end{bmatrix}\begin{bmatrix}x\\y\\1\end{bmatrix}$$

Piquito
  • 29,594