4.0 General equation of second degree

The equation $$a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$$ is the general equation of second degree and represents a pair of straight lines, circle, parabola, ellipse or hyperbola depending on the value of its six constants i.e., $a,\ b,\ c,\ f,\ g,\ h$.

The necessary condition to represent a pair of straight line is $$abc + 2fgh - a{f^2} - b{g^2} - c{h^2} = 0$$ or, \[\left| {\begin{array}{c}a&h&g \\ h&b&f \\ g&f&c \end{array}} \right| = 0\]