Condition for Equation of Second Degree to Represent Two Lines

An equation of the second degree in two variables x and y can be written in the general form:

$$ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 $$

This equation represents a conic section, which could be a parabola, an ellipse, a hyperbola, or a pair of straight lines. In this content, we will focus on the conditions under which this equation represents a pair of straight lines.

Discriminant of a Second Degree Equation

The condition for the above equation to represent a pair of straight lines is related to its discriminant. The discriminant (Δ) of the second-degree equation is given by:

$$ Δ = abc + 2fgh - af^2 - bg^2 - ch^2 $$

For the equation to represent two straight lines, the discriminant must be zero:

$$ Δ = 0 $$

This implies that the conic represented by the equation is degenerate, and in the case of real coefficients, it splits into two lines.

Homogeneous Equation of Second Degree

If the equation does not have the linear terms (terms in x and y only) and the constant term, i.e., it is of the form:

$$ ax^2 + 2hxy + by^2 = 0 $$

Then it always represents a pair of straight lines passing through the origin, provided that the coefficient determinant (also called the characteristic determinant) is zero:

$$ \begin{vmatrix} a & h \ h & b \ \end{vmatrix} = ab - h^2 = 0 $$

Factorization

If the second-degree equation can be factored into two linear factors, then it represents two lines. The factorization is of the form:

$$ (l_1x + m_1y + n_1)(l_2x + m_2y + n_2) = 0 $$

Where each factor represents a line.

Table of Differences and Important Points

Property	Single Line	Pair of Lines
General Form	$ax + by + c = 0$	$ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$
Discriminant	Not applicable	$Δ = abc + 2fgh - af^2 - bg^2 - ch^2$
Condition	Not applicable	$Δ = 0$
Homogeneous Form	$ax + by = 0$	$ax^2 + 2hxy + by^2 = 0$
Characteristic Determinant	Not applicable	$ab - h^2 = 0$
Factorization	Not applicable	$(l_1x + m_1y + n_1)(l_2x + m_2y + n_2) = 0$

Examples

Example 1: Homogeneous Equation

Consider the equation:

$$ x^2 - 2xy + y^2 = 0 $$

Here, $a = 1$, $h = -1$, and $b = 1$. The characteristic determinant is:

$$ ab - h^2 = (1)(1) - (-1)^2 = 1 - 1 = 0 $$

Thus, the equation represents two lines passing through the origin. Factoring the equation gives us:

$$ (x - y)^2 = 0 $$

Which represents the same line (x = y) with a multiplicity of two.

Example 2: Non-Homogeneous Equation

Consider the equation:

$$ x^2 - 4xy + 4y^2 - 8x + 8y - 4 = 0 $$

Here, $a = 1$, $h = -2$, $b = 4$, $g = -4$, $f = 4$, and $c = -4$. The discriminant is:

$$ Δ = (1)(4)(-4) + 2(4)(-2)(-4) - (1)(4)^2 - (4)(-4)^2 - (-2)^2(-4) $$ $$ Δ = -16 + 64 - 16 - 64 + 16 $$ $$ Δ = 0 $$

Thus, the equation represents two lines. Factoring the equation gives us:

$$ (x - 2y - 2)^2 = 0 $$

Which represents the same line (x - 2y = 2) with a multiplicity of two.

In conclusion, the condition for a second-degree equation to represent two lines is that the discriminant must be zero. If the equation is homogeneous, the characteristic determinant must be zero. If the equation can be factored into linear factors, it represents two lines.