Straight Lines

Coordinate geometry unites algebra and Euclidean geometry by expressing geometric points as ordered real number pairs $(x, y)$ and loci as algebraic equations.

• Distance Formula: The distance between $P(x_1, y_1)$ and $Q(x_2, y_2)$ is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• Section Formula: The point $R$ dividing the line segment joining $P(x_1, y_1)$ and $Q(x_2, y_2)$ in the ratio $m : n$ is: $$\text{Internal: } \left(\frac{m x_2 + n x_1}{m + n}, \; \frac{m y_2 + n y_1}{m + n}\right), \quad \text{External: } \left(\frac{m x_2 - n x_1}{m - n}, \; \frac{m y_2 - n y_1}{m - n}\right)$$
• Area of Triangle & Collinearity: Area of $\triangle ABC$ with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ is: $$\Delta = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$$ Points $A, B, C$ are collinear if and only if $\Delta = 0$.

Slope ($m$): If $\theta$ is the inclination of a non-vertical straight line ($\theta \neq 90^\circ$), then the trigonometrical tangent of this angle is defined as the slope or gradient of the line:

• For a line parallel to the $x$-axis (horizontal line), $\theta = 0^\circ \implies m = \tan 0^\circ = \mathbf{0}$.
• For a line parallel to the $y$-axis (vertical line), $\theta = 90^\circ \implies m = \tan 90^\circ$ is undefined (or infinite).
• If $\theta$ is acute ($0^\circ < \theta < 90^\circ$), the slope $m > 0$ (positive slope).
• If $\theta$ is obtuse ($90^\circ < \theta < 180^\circ$), the slope $m < 0$ (negative slope).

Let $P(x_1, y_1)$ and $Q(x_2, y_2)$ with $x_1 \neq x_2$ be two distinct points on a non-vertical line of inclination $\theta$. Constructing a right-angled triangle with horizontal base $(x_2 - x_1)$ and vertical rise $(y_2 - y_1)$:

Collinearity via Slopes: Three points $A, B, C$ are collinear if and only if $\mathbf{\text{Slope}(AB) = \text{Slope}(BC)}$.

Depending on the geometric parameters specified (points, slopes, intercepts), the equation of a straight line can be expressed in various standard forms.

• Line parallel to the $x$-axis: Every point on the line has a constant $y$-coordinate. Equation: $\mathbf{y = k}$ (for $x$-axis itself, $\mathbf{y = 0}$).
• Line parallel to the $y$-axis: Every point on the line has a constant $x$-coordinate. Equation: $\mathbf{x = c}$ (for $y$-axis itself, $\mathbf{x = 0}$).

The equation of a straight line passing through a fixed point $(x_1, y_1)$ and having slope $m$ is derived from the slope definition for any arbitrary point $(x, y)$ on the line:

The equation of a straight line passing through two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ is obtained by substituting $m = \frac{y_2 - y_1}{x_2 - x_1}$ into the point-slope form:

Let a line have slope $m$ and intersect the $y$-axis at $(0, c)$, so that its $y$-intercept is $c$. Substituting $(0, c)$ into point-slope form gives:

If the line makes an $x$-intercept $d$, intersecting the $x$-axis at $(d, 0)$, its equation is $\mathbf{y = m(x - d)}$.

Let a line make non-zero intercepts $a$ on the $x$-axis and $b$ on the $y$-axis, passing through $(a, 0)$ and $(0, b)$. Using two-point form:

When the position of a line is defined through origin-directed perpendicular rays or along-line distances, the normal and parametric forms provide unmatched analytical power.

Let $p$ be the length of the perpendicular (normal) drawn from the origin $O(0, 0)$ to a straight line $L$, and let $\alpha$ be the angle which this normal makes with the positive direction of the $x$-axis ($0 \le \alpha < 360^\circ$ and $p \ge 0$).

The foot of the perpendicular $N$ on the line has polar-to-Cartesian coordinates:

Since the line $L$ is perpendicular to $ON$, its slope is $m_L = -\frac{1}{m_{ON}} = -\frac{1}{\tan \alpha} = -\frac{\cos \alpha}{\sin \alpha}$.

Using the point-slope form with point $N(p\cos\alpha, p\sin\alpha)$:

Let a line pass through a given point $P(x_1, y_1)$ and make an inclination $\theta$ with the positive $x$-axis. If $Q(x, y)$ is any point on the line at an algebraic distance $r$ from $P$:

Every first-degree equation in two variables represents a straight line, and conversely, every straight line can be represented by a linear equation.

When two lines intersect, they form two pairs of vertically opposite angles (one acute, one obtuse), completely determined by their slopes.

Let $L_1$ and $L_2$ be two lines with slopes $m_1$ and $m_2$ and inclinations $\theta_1$ and $\theta_2$ respectively ($\theta_2 > \theta_1$). The angle of intersection is $\theta = \theta_2 - \theta_1$.

• Condition of Parallelism ($\theta = 0^\circ$): $$\tan 0^\circ = 0 \implies m_2 - m_1 = 0 \implies \mathbf{m_1 = m_2}$$ For general equations $A_1 x + B_1 y + C_1 = 0$ and $A_2 x + B_2 y + C_2 = 0$: $$\frac{A_1}{A_2} = \frac{B_1}{B_2} \neq \frac{C_1}{C_2}$$ Equation of any line parallel to $Ax + By + C = 0$ is $\mathbf{Ax + By + k = 0}$.
• Condition of Perpendicularity ($\theta = 90^\circ$): $$\cot 90^\circ = 0 \implies 1 + m_1 m_2 = 0 \implies \mathbf{m_1 m_2 = -1} \quad \left(\text{or } m_2 = -\frac{1}{m_1}\right)$$ For general equations $A_1 x + B_1 y + C_1 = 0$ and $A_2 x + B_2 y + C_2 = 0$: $$A_1 A_2 + B_1 B_2 = 0$$ Equation of any line perpendicular to $Ax + By + C = 0$ is $\mathbf{Bx - Ay + k = 0}$.

The final module addresses spatial metric relations: perpendicular distances, bundles of concurrent lines, and geometric reflection transformations.

If two parallel lines are written with identical coefficients of $x$ and $y$ as $L_1: Ax + By + C_1 = 0$ and $L_2: Ax + By + C_2 = 0$:

Let $L_1 \equiv A_1 x + B_1 y + C_1 = 0$ and $L_2 \equiv A_2 x + B_2 y + C_2 = 0$ be two intersecting lines. The equation of any line passing through their point of intersection is:

where $\lambda \in \mathbb{R}$ is an arbitrary parameter determined by an additional geometric constraint.

Three lines $A_i x + B_i y + C_i = 0$ ($i = 1, 2, 3$) are concurrent (pass through a common single point) if and only if: