Description

COORDINATE GEOMETRY I — Complete Study Guide (003)
Advanced Level Mathematics — Form Five

A comprehensive, exam-ready resource covering everything from the coordinate plane to orthogonal circles — all in one clearly organized document.

📘 What’s Inside
1. Introduction to Coordinate Geometry

Definition (Analytical Geometry) and real-life applications

Key elements: x-axis, y-axis, origin, and the four quadrants

2. The Angle Between Two Lines

Understanding acute vs. obtuse angles (supplementary relationship)

Full formula derivation from slopes to the final tangent formula

Important cases: parallel, perpendicular, and coincident lines

3. Worked Examples — Angles

5 fully worked examples including:

Finding the acute angle between two lines

Finding the tangent of an angle

Equations of lines through a point at a given angle

Lines intersecting at 30° with a known slope

Proof problem (tangent of angle between lines = 2pq/(p² − q²))

4. Perpendicular Distance of a Point from a Line

Visual representation and full formula derivation

The distance formula: r = |ax₁ + by₁ + c| / √(a² + b²)

5. Worked Examples — Distance

Basic distance calculations

Finding unknown values (solving for p or n)

Distance between parallel lines

Distance from the origin

Proof problem (proving m² + 4n² = k²)

6. Locus

Definition and visual concept of a moving point

Table of common loci: circle, perpendicular bisector, angle bisector, ellipse, hyperbola

7. Worked Examples — Locus

Equidistant from two points (perpendicular bisector)

Distance in a given ratio (circle)

Distance from a line

Sum of distances (ellipse derivation)

Locus of an equidistant point

8. Ratio Theorem — Internal Division

Formula derivation using similar triangles

4 fully worked examples, including finding the ratio when a line divides a segment

9. Ratio Theorem — External Division

Formula derivation and alternative formula

4 fully worked examples, including verifying a point and finding unknowns

10. Comparison: Internal vs. External Division

Side-by-side table and quick reference

11. Circles

Definition, key elements, and standard equation table

General equation derivation: x² + y² + 2gx + 2fy + c = 0

Finding centre and radius (two methods)

12. Worked Examples — Circles

Centre and radius from general form (comparison + completing the square)

Equation from centre and radius

Circle touching both axes

Circle through a given point

Circle with a given diameter

Circle through three points (solving simultaneous equations)

13. Tangent to a Circle

Definition and key property (radius ⊥ tangent)

Full equation derivation and standard tangent equation

Worked examples: tangent at a point, and tangents parallel to a given line

14. Normal to a Circle

Definition and key property (normal passes through the centre)

Equation derivation

15. Common Chord & Orthogonal Circles

Finding the common chord and intersection points

Condition for orthogonality (d² = r₁² + r₂²)

Worked examples: proving circles are orthogonal, and finding an orthogonal circle

16. Length of Tangent

Formula and derivation using Pythagoras’ theorem

Fully worked example

17. Summary of All Formulas

11 key formulas in one place: angle, distance, division, circle, tangent, normal, common chord, orthogonality, and tangent length

18. Practice Questions

Five sections: Angles, Perpendicular Distance, Locus, Ratio Theorem, and Circles

19. Quick Reference Card

All essential formulas on one card

✅ Why This Guide Works
Every formula fully derived — not just given, but proven step by step

Proof problems included — build deep understanding, not just memorization

Two methods shown where possible (e.g., comparison vs. completing the square)

Exam-focused — practice questions by topic and a quick reference card

Complete coverage — from the coordinate plane to orthogonal circles in a single document

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