Straight Lines

The inclination of a non-vertical straight line is defined as the angle $\theta$ ($0 \le \theta < \pi$) measured in the anti-clockwise direction from the positive direction of the x-axis to the line. The slope or gradient $m$ of the line is the trigonometric tangent of its inclination: \[ m = \tan\theta \] If a line passes through two distinct coordinate points $P(x_1, y_1)$ and $Q(x_2, y_2)$, its slope is uniquely calculated using the vertical change divided by the horizontal change: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \quad (x_1 \neq x_2) \]

• Parallel Lines: Two non-vertical lines are parallel if and only if their slopes are identically equal ($m_1 = m_2$).
• Perpendicular Lines: Two non-vertical lines are perpendicular if and only if the product of their slopes evaluates to negative unity: \[ m_1 \cdot m_2 = -1 \implies m_2 = -\frac{1}{m_1} \]

• Slope-Intercept Form: $y = mx + c$ (where $c$ is the y-intercept)
• Point-Slope Form: $y - y_1 = m(x - x_1)$ (passes through $P(x_1, y_1)$)
• Two-Point Form: $y - y_1 = \left(\frac{y_2 - y_1}{x_2 - x_1}\right)(x - x_1)$
• Intercept Form: $\frac{x}{a} + \frac{y}{b} = 1$ (where $a$ and $b$ are the x and y intercepts)
• Normal Form: $x \cos\alpha + y \sin\alpha = p$ (where $p$ is the perpendicular distance from the origin and $\alpha$ is the inclination angle of that perpendicular vector)
• Symmetric / Parametric Form: Tracks coordinates step-wise using a directional distance parameter $r$ from a baseline point $P(x_1, y_1)$ along an inclination angle $\theta$: \[ \frac{x - x_1}{\cos\theta} = \frac{y - y_1}{\sin\theta} = r \implies x = x_1 + r\cos\theta, \quad y = y_1 + r\sin\theta \]

• Distance of a Point from a Line: The perpendicular distance $d$ from a point $P(x_1, y_1)$ to the general line $Ax + By + C = 0$ is: \[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
• Distance Between Parallel Lines: Two parallel lines share identical leading coefficients when normalized, written as $L_1: Ax + By + C_1 = 0$ and $L_2: Ax + By + C_2 = 0$. The perpendicular distance between them is the absolute difference of their constants over the norm coefficient: \[ d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} \]

• Foot of the Perpendicular: Let $H(h, k)$ represent the coordinates of the foot of the perpendicular dropped from a point $P(x_1, y_1)$ onto the line $Ax + By + C = 0$. The coordinate shifts are calculated using the ratio formula: \[ \frac{h - x_1}{A} = \frac{k - y_1}{B} = -\frac{Ax_1 + By_1 + C}{A^2 + B^2} \]
• Reflection (Image) of a Point: Let $I(x_2, y_2)$ represent the coordinates of the mirror reflection image of a point $P(x_1, y_1)$ across the line mirror $Ax + By + C = 0$. Since the line acts as the perpendicular bisector connecting the object $P$ and its image $I$, the transformation ratio doubles its scalar displacement factor: \[ \frac{x_2 - x_1}{A} = \frac{y_2 - y_1}{B} = -2\left(\frac{Ax_1 + By_1 + C}{A^2 + B^2}\right) \]

A set of three or more distinct straight lines is said to be concurrent if they all pass through the exact same point of intersection. Three lines given by $L_i: A_i x + B_i y + C_i = 0$ (where $i = 1, 2, 3$) are concurrent if and only if the determinant of their coefficients evaluates to zero: \[ \begin{vmatrix} A_1 & B_1 & C_1 \\ A_2 & B_2 & C_2 \\ A_3 & B_3 & C_3 \end{vmatrix} = 0 \]

The infinite set of lines passing through the intersection of two baseline lines $L_1 = 0$ and $L_2 = 0$ is defined as a family of straight lines. Any line belonging to this family can be modeled using the linear combination formula: \[ L_1 + \lambda L_2 = 0 \implies (A_1 x + B_1 y + C_1) + \lambda(A_2 x + B_2 y + C_2) = 0 \] where $\lambda \in \mathbb{R}$ is a variable scalar parameter. This formula allows you to find a specific line from the family by applying a given geometric constraint to solve for $\lambda$.

• Coincident Lines: The lines are coincident (or parallel) if $h^2 - ab = 0 \implies h^2 = ab$.
• Perpendicular Lines: The lines are perpendicular if the sum of the squared coefficients vanishes: $a + b = 0$.

The general second-degree equation in terms of $x$ and $y$ incorporates linear and constant terms along with the quadratic terms: \[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \] This general equation represents a pair of straight lines if and only if the determinant of its complete coefficient matrix vanishes. This condition is represented by the discriminant equation $\Delta = 0$: \[ \Delta = abc + 2fgh - af^2 - bg^2 - ch^2 = 0 \] Additionally, to represent real lines, the coefficients must satisfy $h^2 \ge ab$.

If $\Delta = 0$, the general equation can be factored into two distinct linear equations representing lines that intersect at an interior point away from the origin. The coordinates of this intersection point can be found by taking the partial derivatives of the general equation with respect to $x$ and $y$ and solving the resulting linear system: \[ \frac{\partial f}{\partial x} = 2ax + 2hy + 2g = 0 \quad \text{and} \quad \frac{\partial f}{\partial y} = 2hx + 2by + 2f = 0 \]