8.1 Coordinates, Slopes, Straight Line Equations & Distances

Coordinates, Slopes, Straight Line Equations & Distances

Analytic geometry bridges algebraic formulation and geometric spatial intuition, serving as one of the highest-yield topics in the Intermediate Mathematics syllabus and the University of Engineering and Technology (UET) ECAT. Success on this paper requires students to transcend tedious manual graphing by deploying direct algebraic transformations, parameter identifications, and ratio shortcuts.

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1. Cartesian Coordinates, Distance Metric, and Point Division

A point $P$ in the Cartesian coordinate plane $\mathbb{R}^2$ is specified by an ordered pair of real numbers $(x, y)$, where $x$ denotes the directed distance along the horizontal abscissa and $y$ denotes the directed distance along the vertical ordinate.

The Euclidean Distance Metric

For two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$, the straight-line distance $d = |P_1 P_2|$ is derived directly from the Pythagorean theorem on the right-angled triangle formed with horizontal leg $|x_2 - x_1|$ and vertical leg $|y_2 - y_1|$:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Three points $A, B,$ and $C$ are collinear if and only if the sum of the distances between two adjacent pairs equals the distance between the extremal pair (e.g., $AB + BC = AC$), or equivalently, if the area of the triangle formed by them is zero.

The Section Formula

Let $P(x, y)$ divide the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m_1 : m_2$:

1. Internal Division: When $P$ lies strictly between $A$ and $B$:

$P(x, y) = \left( \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \; \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \right)$

1. External Division: When $P$ lies on the line extension beyond $A$ or $B$ ($m_1 \neq m_2$):

$P'(x, y) = \left( \frac{m_1 x_2 - m_2 x_1}{m_1 - m_2}, \; \frac{m_1 y_2 - m_2 y_1}{m_1 - m_2} \right)$

1. Midpoint Formula ($m_1 = m_2 = 1$):

$M(x, y) = \left( \frac{x_1 + x_2}{2}, \; \frac{y_1 + y_2}{2} \right)$

ECAT Ratio Shortcut: If a point $P(x, y)$ divides the segment $AB$ in the ratio $k : 1$, setting $k = \frac{m_1}{m_2}$ yields $x = \frac{k x_2 + x_1}{k + 1}$ and $y = \frac{k y_2 + y_1}{k + 1}$. If $k > 0$, the division is internal; if $k < 0$, the division is external.

Triangle Centers & Area

• Centroid ($G$): The concurrency point of the three medians of a triangle with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ divides each median in a $2:1$ ratio from vertex to midpoint:

$G = \left( \frac{x_1 + x_2 + x_3}{3}, \; \frac{y_1 + y_2 + y_3}{3} \right)$

• Area of a Triangle (Determinant Form): The signed area $\Delta$ of the triangle formed by vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ is:

$\Delta = \frac{1}{2} \left| \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix} \right| = \frac{1}{2} \bigl| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \bigr|$

Collinearity of three points is verified algebraically if and only if this $3 \times 3$ determinant equals zero.

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2. Inclination, Slope, and Relative Orientations of Lines

The angle of inclination $\alpha$ of a straight line is the directed angle measured counterclockwise from the positive $x$-axis to the line, with domain $\alpha \in [0, \pi)$ or $0^\circ \le \alpha < 180^\circ$.

Slope (Gradient) of a Line

The slope $m$ is the tangent of the inclination angle:

$m = \tan\alpha = \frac{y_2 - y_1}{x_2 - x_1} \quad (x_1 \neq x_2)$

• Horizontal lines have $\alpha = 0^\circ \implies m = \tan 0^\circ = 0$.
• Vertical lines have $\alpha = 90^\circ \implies m = \tan 90^\circ$ is undefined (approaching $\pm\infty$).
• Lines inclined at an acute angle have $m > 0$; lines inclined at an obtuse angle have $m < 0$.

Angle Between Two Intersecting Lines

Let two non-vertical lines $L_1$ and $L_2$ have slopes $m_1$ and $m_2$, with inclinations $\alpha_1$ and $\alpha_2$. The acute angle $\theta$ between them satisfies:

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

Parallelism and Perpendicularity Criteria

1. Parallel Lines ($L_1 \parallel L_2$): Two lines are parallel if and only if their inclinations are equal, meaning their slopes are identical:

$m_1 = m_2$

1. Perpendicular Lines ($L_1 \perp L_2$): Two lines are mutually orthogonal if and only if $\theta = 90^\circ$, meaning the denominator vanishes:

$1 + m_1 m_2 = 0 \iff m_1 m_2 = -1 \iff m_2 = -\frac{1}{m_1}$

ECAT Trap Alert: The relation $m_1 m_2 = -1$ holds only when neither line is vertical. If one line is vertical ($x = k$, undefined slope), any perpendicular line is strictly horizontal ($y = c$, slope $m = 0$).

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3. Standard Algebraic Forms of Straight Lines

Conversion of General Form to Normal Form

To transform $Ax + By + C = 0$ into the normal form $x\cos\alpha + y\sin\alpha = p$, rearrange terms so the constant $p \ge 0$ appears on the right side: $-Ax - By = C$. Dividing both sides by $\pm\sqrt{A^2 + B^2}$ yields:

$\left( -\frac{A}{\pm\sqrt{A^2+B^2}} \right)x + \left( -\frac{B}{\pm\sqrt{A^2+B^2}} \right)y = \frac{C}{\pm\sqrt{A^2+B^2}}$

To guarantee $p > 0$, choose the sign of the radical to match the sign of the constant $C$, so that $p = \frac{|C|}{\sqrt{A^2+B^2}}$. Example: $3x - 4y + 10 = 0$ gives $-3x + 4y = 10$; dividing by $+5$ gives $-\frac{3}{5}x + \frac{4}{5}y = 2$, so $p = 2$, $\cos\alpha = -\frac{3}{5}$ and $\sin\alpha = \frac{4}{5}$.

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4. Perpendicular Distances and Parallel Line Separation

Distance from an Arbitrary Point to a Line

The perpendicular distance $d$ from a given point $P(x_0, y_0)$ to the straight line $L \equiv Ax + By + C = 0$ is:

$d = \frac{|A x_0 + B y_0 + C|}{\sqrt{A^2 + B^2}}$

For the special case where $P$ is the origin $(0, 0)$:

$d_0 = \frac{|C|}{\sqrt{A^2 + B^2}}$

Distance Between Two Parallel Lines

Let two parallel lines be expressed with identical linear coefficients $A$ and $B$:

$L_1 \equiv Ax + By + C_1 = 0 \quad \text{and} \quad L_2 \equiv Ax + By + C_2 = 0$

The perpendicular separation distance $d$ between them is:

$d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$

ECAT Trap Alert: Before applying $d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$, normalize both linear equations so the coefficients of $x$ and $y$ are completely identical. For example, if given $3x - 4y + 5 = 0$ and $6x - 8y - 10 = 0$, divide the second equation by 2 to yield $3x - 4y - 5 = 0$ before computing $d = \frac{|5 - (-5)|}{\sqrt{3^2 + (-4)^2}} = \frac{10}{5} = 2$.

Relative Positions of Points with Respect to a Line

Two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ lie on the same side of $Ax + By + C = 0$ if $(Ax_1 + By_1 + C)$ and $(Ax_2 + By_2 + C)$ share the same sign. They lie on opposite sides if these evaluations have opposite algebraic signs.

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5. Homogeneous Second-Degree Equation & Pairs of Straight Lines

A second-degree homogeneous algebraic equation in $x$ and $y$ has the standard form:

$ax^2 + 2hxy + by^2 = 0$

Because every term is quadratic, this equation represents a pair of straight lines passing through the origin $(0, 0)$. Dividing by $x^2$ ($x \neq 0$) and setting $m = \frac{y}{x}$ gives:

$b m^2 + 2h m + a = 0$

Let $m_1$ and $m_2$ be the roots of this quadratic equation, corresponding to the slopes of the two lines $y = m_1 x$ and $y = m_2 x$. By Vieta's formulas:

$m_1 + m_2 = -\frac{2h}{b}, \qquad m_1 m_2 = \frac{a}{b}$

Angle Between the Pair of Lines

The angle $\theta$ between the two lines represented by $ax^2 + 2hxy + by^2 = 0$ is:

$\tan\theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| = \frac{\sqrt{(m_1 + m_2)^2 - 4m_1 m_2}}{1 + m_1 m_2} = \frac{\sqrt{\left(-\frac{2h}{b}\right)^2 - 4\left(\frac{a}{b}\right)}}{1 + \frac{a}{b}} = \frac{2\sqrt{h^2 - ab}}{a + b}$

Geometric Conditions for the Pair of Lines

1. Real and Distinct Lines: $h^2 - ab > 0$ ($h^2 > ab$).
2. Real and Coincident Lines: $h^2 - ab = 0 \iff h^2 = ab$.
3. Imaginary Lines (intersecting only at the origin): $h^2 - ab < 0$.
4. Orthogonal (Mutually Perpendicular) Lines: The lines are perpendicular if $\tan\theta \to \infty$, which occurs when the denominator equals zero:

$a + b = 0 \iff \text{Coefficient of } x^2 + \text{Coefficient of } y^2 = 0$

ECAT Speed Trick: To immediately check whether a homogeneous second-degree equation represents perpendicular lines, inspect $a + b$. If $a + b = 0$ (e.g., $3x^2 + 7xy - 3y^2 = 0$), the lines are mutually perpendicular regardless of the value of $h$.

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6. General Second-Degree Equation Representing Two Lines

The general non-homogeneous second-degree equation:

$ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$

represents a pair of coplanar straight lines if and only if the determinant $\Delta$ vanishes:

$\Delta = \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} = abc + 2fgh - af^2 - bg^2 - ch^2 = 0$

If $\Delta = 0$, the angle between these lines is still governed by $\tan\theta = \frac{2\sqrt{h^2 - ab}}{a + b}$, and they are mutually perpendicular if $a + b = 0$ and parallel if $h^2 = ab$.