The Straight Line

The Straight Line

In analytic geometry, a line is defined as the locus of a point moving in a constant direction. It is a straight one-dimensional figure having no thickness and extending infinitely in both directions.

General Equation of a Line

The general equation of a straight line is a linear equation in two variables, $x$ and $y$:

General Equation

$Ax + By + C = 0$

Where $A$ and $B$ are not both zero.

Forms of the Equation of a Line

Depending on the given information, the equation of a line can be written in several forms.

1. Point-Slope Form

Given a point $(x_1, y_1)$ and a slope $m$: $y - y_1 = m(x - x_1)$

2. Slope-Intercept Form

Given a slope $m$ and y-intercept $b$: $y = mx + b$

3. Two-Point Form

Given two points $(x_1, y_1)$ and $(x_2, y_2)$: $y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1)$

4. Intercept Form

Given x-intercept $a$ and y-intercept $b$: $\frac{x}{a} + \frac{y}{b} = 1$

5. Normal Form

Given the perpendicular distance $p$ from the origin to the line and the angle $\omega$ that the perpendicular makes with the positive x-axis: $x \cos \omega + y \sin \omega = p$

Parallel and Perpendicular Lines

Two lines with slopes $m_1$ and $m_2$ are:

• Parallel if their slopes are equal ($m_1 = m_2$).
• Perpendicular if the product of their slopes is -1 ($m_1 m_2 = -1$).

Distance from a Point to a Line

The perpendicular distance $d$ from a point $(x_1, y_1)$ to the line $Ax + By + C = 0$ is:

Distance from Point to Line

$d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$

Angle Between Two Lines

If $\theta$ is the angle between two lines having slopes $m_1$ and $m_2$:

Angle Between Lines

$\tan \theta = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right|$

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Solved Problems