Coordinate Geometry - Straight Lines (Various Forms and General Equation)

Inclination and Slope: The inclination of a line is the angle $\theta$ ($0 \le \theta < 180^{\circ}$) which the line makes with the positive direction of the x-axis, measured in the anticlockwise direction. Visually, a line leaning to the right has an acute angle and positive slope, while a line leaning to the left has an obtuse angle and negative slope. The slope $m$ is defined as $m = \tan \theta$.

Slope-Intercept Form: A line with slope $m$ and y-intercept $c$ is represented by $y = mx + c$. Graphically, the y-intercept $c$ is the vertical distance from the origin where the line crosses the y-axis at point $(0, c)$. If the line passes through the origin, $c = 0$ and the equation simplifies to $y = mx$.

Point-Slope and Two-Point Form: The Point-Slope form $y - y_1 = m(x - x_1)$ represents a line passing through a fixed point $(x_1, y_1)$ with a specific tilt $m$. When two points $(x_1, y_1)$ and $(x_2, y_2)$ are given, the slope is first calculated as the 'rise over run' ratio, leading to the Two-Point form which defines the unique straight path connecting them.

Intercept Form: The equation $\frac{x}{a} + \frac{y}{b} = 1$ represents a line that cuts the x-axis at $(a, 0)$ and the y-axis at $(0, b)$. Visually, $a$ and $b$ represent the signed lengths of the segments intercepted on the axes from the origin. This form is particularly useful for sketching lines quickly by marking two points on the axes.

Normal Form: A line can be defined by the length of the perpendicular (normal) drawn from the origin to the line, denoted by $p$, and the angle $\alpha$ that this normal makes with the positive x-axis. The equation is $x \cos \alpha + y \sin \alpha = p$. Note that $p$ is always positive as it represents distance.

General Equation of a Line: Any first-degree equation in $x$ and $y$ of the form $Ax + By + C = 0$ (where $A$ and $B$ are not both zero) represents a straight line. By rearranging this into slope-intercept form, we see the slope is $-\frac{A}{B}$ and the y-intercept is $-\frac{C}{B}$.

Parallel and Perpendicular Lines: Two non-vertical lines are parallel if and only if their slopes are equal ($m_1 = m_2$), meaning they have the same inclination and never meet. They are perpendicular if and only if the product of their slopes is $-1$ ($m_1 \cdot m_2 = -1$), which occurs when the lines intersect at a $90^{\circ}$ angle.