Coordinate Geometry

Overview

Coordinate Geometry (also called Analytic Geometry) bridges algebra and geometry by representing geometric figures using numbers and equations on a plane. For KAR TET Paper II Mathematics, this topic carries significant weight as it tests both conceptual understanding and computational accuracy.

Students must master three core areas: plotting and interpreting points on the Cartesian plane, calculating distances between points, and finding coordinates of points that divide a line segment in a given ratio. These concepts form the foundation for higher geometry and appear frequently in classroom teaching scenarios that TET candidates must handle.

The topic connects directly to real-world applications—mapping locations, calculating shortest paths, and dividing land or resources proportionally—making it ideal for activity-based teaching that TET pedagogy emphasises.

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Key Concepts

• **Cartesian Plane**: A plane formed by two perpendicular number lines—the horizontal x-axis and vertical y-axis—intersecting at the origin O(0, 0).

• **Coordinates of a Point**: Any point P on the plane is represented as an ordered pair (x, y), where x is the perpendicular distance from the y-axis (abscissa) and y is the perpendicular distance from the x-axis (ordinate).

• **Four Quadrants**: The axes divide the plane into four regions—Quadrant I (+, +), Quadrant II (–, +), Quadrant III (–, –), and Quadrant IV (+, –).

• **Distance Formula**: Derived from the Pythagorean theorem; gives the length of the line segment joining two points.

• **Section Formula**: Determines the coordinates of a point dividing a line segment internally in a given ratio m:n.

• **Midpoint**: A special case of section formula where the ratio is 1:1.

• **Collinearity**: Three points are collinear if the area of the triangle formed by them equals zero, or equivalently, if the sum of two smaller distances equals the third.

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Formulas / Key Facts

**Distance Formula** For points A(x₁, y₁) and B(x₂, y₂): Distance AB = √[(x₂ – x₁)² + (y₂ – y₁)²]

**Section Formula (Internal Division)** Point P dividing the line segment joining A(x₁, y₁) and B(x₂, y₂) internally in ratio m:n: P = [(mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)]

**Midpoint Formula** When m:n = 1:1: Midpoint M = [(x₁ + x₂)/2, (y₁ + y₂)/2]

**Section Formula (External Division)** Point P dividing externally in ratio m:n: P = [(mx₂ – nx₁)/(m – n), (my₂ – ny₁)/(m – n)]

**Distance from Origin** For point P(x, y): Distance from origin = √(x² + y²)

**Key Sign Conventions**

• Points on x-axis have y = 0
• Points on y-axis have x = 0
• Origin is (0, 0)

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Worked Examples

**Example 1: Distance Formula** *Find the distance between A(3, 4) and B(–1, 1).*

Step 1: Identify coordinates x₁ = 3, y₁ = 4, x₂ = –1, y₂ = 1

Step 2: Apply formula AB = √[(–1 – 3)² + (1 – 4)²] AB = √[(–4)² + (–3)²] AB = √[16 + 9] AB = √25 = **5 units**

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**Example 2: Section Formula** *Find the coordinates of point P that divides the line segment joining A(2, –3) and B(7, 2) in the ratio 2:3 internally.*

Step 1: Identify values m = 2, n = 3, x₁ = 2, y₁ = –3, x₂ = 7, y₂ = 2

Step 2: Apply section formula x-coordinate = (2 × 7 + 3 × 2)/(2 + 3) = (14 + 6)/5 = 20/5 = 4 y-coordinate = (2 × 2 + 3 × (–3))/(2 + 3) = (4 – 9)/5 = –5/5 = –1

**P = (4, –1)**

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**Example 3: Midpoint and Verification** *Find the midpoint of the segment joining P(–2, 8) and Q(6, –4). Verify using the distance formula.*

Step 1: Find midpoint M M = [(–2 + 6)/2, (8 + (–4))/2] M = [4/2, 4/2] = **(2, 2)**

Step 2: Verify (PM should equal MQ) PM = √[(2 – (–2))² + (2 – 8)²] = √[16 + 36] = √52 MQ = √[(6 – 2)² + (–4 – 2)²] = √[16 + 36] = √52

PM = MQ ✓ (M is indeed the midpoint)

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Common Mistakes

**Wrong thinking**: Subtracting coordinates in wrong order and getting confused by negative signs. **Correct fix**: The order doesn't affect the final answer since the difference is squared. However, maintain consistency: always do (x₂ – x₁) and (y₂ – y₁).

**Wrong thinking**: In section formula, placing m with the wrong point's coordinates. **Correct fix**: Remember "m goes with the farther point"—if P divides AB in ratio m:n, then m multiplies B's coordinates (the point towards which we measure).

**Wrong thinking**: Forgetting to take the square root in the distance formula, giving the answer as the sum of squares. **Correct fix**: Distance = √(sum of squares). Always complete the final step.

**Wrong thinking**: Confusing internal and external division formulas. **Correct fix**: Internal division uses (m + n) in the denominator; external division uses (m – n). For most school-level problems, internal division is asked.

**Wrong thinking**: Assuming midpoint divides in ratio 1:2 instead of 1:1. **Correct fix**: Midpoint means equal parts—ratio is always 1:1.

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Quick Reference

• **Quadrant signs**: I (+,+) → II (–,+) → III (–,–) → IV (+,–) [anticlockwise from top-right]

• **Distance formula**: √[(x₂ – x₁)² + (y₂ – y₁)²]

• **Section formula (internal)**: x = (mx₂ + nx₁)/(m+n); y = (my₂ + ny₁)/(m+n)

• **Midpoint**: Average of x-coordinates, average of y-coordinates

• **Points on axes**: On x-axis → y = 0; On y-axis → x = 0

• **Collinearity check**: If AB + BC = AC, then A, B, C are collinear