Pair of Straight Lines

In geometry, the concept of a pair of straight lines arises when we consider two lines that may or may not intersect at a point in a plane. When these lines are represented algebraically, they can be expressed as a single quadratic equation in two variables. Understanding this concept is crucial for solving various problems in coordinate geometry.

General Equation of a Pair of Straight Lines

The general second-degree equation in two variables x and y is given by:

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

This equation represents a pair of straight lines if and only if the determinant of the coefficients of $x^2$, $xy$, and $y^2$ is zero, i.e.,

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

When this condition is satisfied, the equation can be factored into two linear factors, each representing a straight line.

Homogeneous Equation of a Pair of Straight Lines

If the general equation does not have the terms $2gx$, $2fy$, and $c$, it is called a homogeneous equation of the second degree:

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

This equation always represents a pair of straight lines passing through the origin.

Angle Between the Pair of Straight Lines

The angle $\theta$ between the two lines represented by the equation $ax^2 + 2hxy + by^2 = 0$ is given by:

$$ \tan\theta = \frac{2\sqrt{h^2 - ab}}{a + b} $$

If $a + b = 0$, the lines are perpendicular.

Point of Intersection

For the general equation $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$, if it represents a pair of straight lines, the point of intersection of these lines is given by:

$$ (x, y) = \left(\frac{-g}{a}, \frac{-f}{b}\right) $$

Differences and Important Points

Feature	Homogeneous Equation	General Equation
Form	$ax^2 + 2hxy + by^2 = 0$	$ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$
Represents	Always a pair of lines through the origin	A pair of lines if $ab = h^2$
Angle Between Lines	Given by $\tan\theta = \frac{2\sqrt{h^2 - ab}}{a + b}$	Same formula applies
Point of Intersection	Always at the origin (0, 0)	Given by $\left(\frac{-g}{a}, \frac{-f}{b}\right)$ if lines intersect

Examples

Example 1: Homogeneous Equation

Consider the equation $x^2 - 2xy + y^2 = 0$. This is a homogeneous equation and can be factored as:

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

This represents a pair of coincident straight lines, both being $x - y = 0$.

Example 2: General Equation

The equation $x^2 - 4xy + 4y^2 - 8x + 16y - 16 = 0$ can be rewritten as:

$$ (x - 2y)^2 - 8(x - 2y) + 16 = 0 $$

Factoring, we get:

$$ (x - 2y - 4)^2 = 0 $$

This represents a pair of coincident straight lines, both being $x - 2y - 4 = 0$.

Example 3: Angle Between Lines

For the equation $x^2 - 4xy + y^2 = 0$, we have $a = 1$, $b = 1$, and $h = -2$. The angle between the lines is:

$$ \tan\theta = \frac{2\sqrt{(-2)^2 - (1)(1)}}{1 + 1} = \frac{2\sqrt{3}}{2} = \sqrt{3} $$

Thus, $\theta = 60^\circ$.

Example 4: Point of Intersection

For the equation $x^2 + 2xy + y^2 - 6x - 6y + 9 = 0$, we can rewrite it as:

$$ (x + y - 3)^2 = 0 $$

This represents a pair of coincident straight lines, both being $x + y - 3 = 0$. The point of intersection is $(3, 0)$.

Understanding the concept of a pair of straight lines is essential for solving complex problems in coordinate geometry, especially when dealing with conic sections and systems of equations.