Introduction
In analytic geometry, the general equation of the second degree in two variables $x$ and $y$ is given by:
$ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$
While this equation often represents conic sections (like circles, ellipses, parabolas, or hyperbolas), it can also represent a pair of straight lines. In this post, we will derive the specific condition under which this happens.
The Condition for a Pair of Straight Lines
For the general second-degree equation to represent a pair of straight lines, the equation must be factorable into two linear factors of the form $(l_1x + m_1y + n_1)(l_2x + m_2y + n_2) = 0$.
The algebraic condition for this to be possible is that the determinant of the coefficients must be zero. This is expressed as:
$\begin{vmatrix} a & h & g \ h & b & f \ g & f & c \end{vmatrix} = 0$
Expanding the Determinant
To find the expanded form of this condition, we evaluate the determinant:
$a(bc - f^2) - h(hc - fg) + g(hf - bg) = 0$
Expanding this, we get the standard form:
$abc + 2fgh - af^2 - bg^2 - ch^2 = 0$
Intuition and Conclusion
- Geometric Meaning: If the discriminant of the quadratic form is zero, the conic represented by the equation has "degenerated" into two lines.
- Practical Application: If you are given a specific quadratic equation and need to determine if it describes two lines, simply calculate the determinant value $\Delta = abc + 2fgh - af^2 - bg^2 - ch^2$. If $\Delta = 0$, the equation represents a pair of lines.
- Significance: This is a fundamental concept in coordinate geometry, often used to study the intersection of lines and the properties of degenerate conics.