1.6 Coordinate Geometry: Lines, Distance, Midpoints & Conic Sections

Coordinate Geometry: Lines, Distance, Midpoints & Conic Sections

Coordinate geometry bridges algebra and plane geometry, representing one of the most heavily tested content domains on the Ateneo College Entrance Test (ACET) Mathematics subtest. Success on the ACET requires rapid execution of distance and midpoint formulas, linear slope derivations, perpendicular distance calculations from points to lines, Shoelace polygon area determinations, and structural identification of conic sections (circles, parabolas, ellipses, hyperbolas).

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1. Fundamental Cartesian Formulas

Given two arbitrary points in the Cartesian plane $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$:

Primary Metric Formulas

• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
• Midpoint Formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
• Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1} \quad (x_1 \neq x_2)$
• Section Formula (Internal Partitioning ratio $m:n$): $P = \left( \frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n} \right)$
• Centroid of a Triangle $(x_1, y_1), (x_2, y_2), (x_3, y_3)$: $C = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)$

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2. Linear Equations, Parallelism & Perpendicular Distance

Linear Forms

1. Slope-Intercept Form: $y = mx + b$ ($m$ is slope, $b$ is y-intercept)
2. Point-Slope Form: $y - y_1 = m(x - x_1)$
3. General Standard Form: $Ax + By + C = 0$ (where slope $m = -A/B$, y-intercept = $-C/B$)

Parallel & Perpendicular Line Relationships

• Parallel Lines ($L_1 \parallel L_2$): Slopes are equal ($m_1 = m_2$). Lines never intersect unless identical.
• Perpendicular Lines ($L_1 \perp L_2$): Slopes are negative reciprocals ($m_1 \cdot m_2 = -1 \implies m_2 = -\frac{1}{m_1}$).

Perpendicular Distance from Point $(x_0, y_0)$ to Line $Ax + By + C = 0$

The shortest distance from an external point $(x_0, y_0)$ to a straight line expressed in general form $Ax + By + C = 0$ is:

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

Shoelace Formula for Polygon Area

To calculate the area of a polygon with ordered vertices $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$:

$$\text{Area} = \frac{1}{2} \left| (x_1 y_2 + x_2 y_3 + \dots + x_n y_1) - (y_1 x_2 + y_2 x_3 + \dots + y_n x_1) \right|$$

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3. Conic Sections: Canonical Equations & Structural Properties

A conic section is formed by the intersection of a plane and a double-napped cone. In general form without xy-rotation ($B=0$), conics follow:

$$A x^2 + C y^2 + D x + E y + F = 0$$

Conic Section Taxonomy & Properties Table

Identifying Conics via Coefficients ($A x^2 + C y^2 + D x + E y + F = 0$)

• Parabola: $A = 0$ OR $C = 0$ (contains exactly one squared term).
• Circle: $A = C \neq 0$ (squared terms have identical non-zero coefficients with the same sign).
• Ellipse: $A \neq C$, but $A \cdot C > 0$ (squared terms have different positive coefficients of the same sign).
• Hyperbola: $A \cdot C < 0$ (squared terms have coefficients of opposite signs).

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Worked Step-by-Step ACET Exam Problems

Problem 1: Perpendicular Distance from Point to Line

Find the exact perpendicular distance from point $P(3, -2)$ to the line $5x - 12y + 7 = 0$.

Step 1: Extract general line coefficients and point coordinates. $A = 5$, $B = -12$, $C = 7$, $x_0 = 3$, $y_0 = -2$.

Step 2: Substitute into point-to-line distance formula. $d = \frac{|A x_0 + B y_0 + C|}{\sqrt{A^2 + B^2}} = \frac{|5(3) + (-12)(-2) + 7|}{\sqrt{5^2 + (-12)^2}}$

Step 3: Evaluate absolute numerator and radical denominator.

• Numerator: $|15 + 24 + 7| = |46| = 46$
• Denominator: $\sqrt{25 + 144} = \sqrt{169} = 13$
• Final Distance: $d = \frac{46}{13} \approx 3.538$.

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Problem 2: Completing the Square to Analyze a Circle Equation

Determine the center $(h, k)$ and radius $r$ of the circle defined by $x^2 + y^2 - 6x + 8y - 11 = 0$.

Step 1: Group x and y variables and isolate constant term. $(x^2 - 6x) + (y^2 + 8y) = 11$.

Step 2: Complete the square for both variable groups.

• For $x$: $\left(\frac{-6}{2}\right)^2 = (-3)^2 = 9$.
• For $y$: $\left(\frac{8}{2}\right)^2 = (4)^2 = 16$. Add $9 + 16 = 25$ to both sides of the equation: $(x^2 - 6x + 9) + (y^2 + 8y + 16) = 11 + 9 + 16$.

Step 3: Factor squared terms into standard canonical form. $(x - 3)^2 + (y + 4)^2 = 36$.

Step 4: Read center coordinates and radius.

• Center $(h, k) = (3, -4)$.
• Radius $r = \sqrt{36} = 6$.

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ACET Speed Tactics & Traps Summary