Chapter 4 Pair of Straight Lines
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Project on Pair of Straight Lines
In mathematics, a pair of straight lines refers to two lines that can be represented by a single second-degree polynomial equation in the form of a conic section. Specifically, this involves equations that can be factored into the product of two linear equations. This concept is especially relevant in coordinate geometry and algebraic geometry. Here's a detailed explanation:
General Form
A second-degree polynomial equation in two variables and can be written as:
This equation represents a conic section, which could be a circle, ellipse, parabola, hyperbola, or a pair of straight lines.
Pair of Straight Lines
A pair of straight lines is a specific case where the conic section degenerates into two lines. This happens when the quadratic equation can be factored into two linear factors:
Conditions for Pair of Straight Lines
For the given quadratic equation to represent a pair of straight lines, the determinant of the coefficient matrix must be zero:
a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} = 0 \] This determinant condition ensures that the quadratic form can be factored into linear components. ### Factoring the Equation If the quadratic equation represents a pair of straight lines, it can be factored as: \[ (lx + my + n)(px + qy + r) = 0 \] Where \(lx + my + n = 0\) and \(px + qy + r = 0\) are the equations of the two lines. ### Angle Between the Lines The angle \(\theta\) between the two lines \(lx + my + n = 0\) and \(px + qy + r = 0\) can be found using the formula: \[ \tan\theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right| \] Where \(m_1\) and \(m_2\) are the slopes of the two lines. For the quadratic equation \(ax^2 + 2hxy + by^2 = 0\), the slopes of the lines can be found from the quadratic equation: \[ y = m_1 x \quad \text{and} \quad y = m_2 x \] Where \(m_1\) and \(m_2\) are the roots of the quadratic equation: \[ b m^2 + 2h m + a = 0 \] ### Example Consider the equation: \[ x^2 - 4xy + 3y^2 = 0 \] To factorize it, we solve the quadratic equation for \(m\): \[ 3m^2 - 4m + 1 = 0 \] The roots are: \[ m_1 = 1 \quad \text{and} \quad m_2 = \frac{1}{3} \] So, the lines are: \[ y = x \quad \text{and} \quad y = \frac{1}{3}x \] Thus, the original equation represents the pair of lines: \[ (x - y)(3x - y) = 0 \] ### Conclusion The concept of a pair of straight lines in mathematics involves understanding how a second-degree polynomial equation can represent two intersecting lines. By examining the determinant of the coefficient matrix and factoring the quadratic equation, we can identify and analyze the lines represented by the equation.