📐 Mathematics 1 - Week 2: Coordinate Geometry & Linear Models

“In coordinate geometry, every point tells a story. Learn to read the coordinates.”

📋 Week 2 Overview

Week 2 builds on algebraic foundations to explore geometry through coordinates:

Lines & Equations - Slope, intercepts, parallel/perpendicular lines
Distance & Optimization - Shortest paths, minimum distances
Linear Models - Cost analysis, optimization problems
Data Analysis - Linear regression, curve fitting
Applications - Real-world problem solving

📏 Pattern 1: Lines and Their Equations

📖 Concept Explanation

Standard Forms:

Slope-intercept: y = mx + c (m = slope, c = y-intercept)
Two-point form: $\frac{y - y _{1}}{y _{2} - y _{1}} = \frac{x - x _{1}}{x _{2} - x _{1}}$
General form: ax + by + c = 0
Intercept form: $\frac{x}{a} + \frac{y}{b} = 1$

Key Relationships:

Parallel lines: same slope (m₁ = m₂)
Perpendicular lines: m₁ × m₂ = -1
Distance from point (x₀,y₀) to line ax + by + c = 0: $\frac{∣ a x _{0} + b y _{0} + c ∣}{a ^{2} + b ^{2}}$

🧠 Mental Algorithm

Identify given information (points, slopes, intercepts)
Choose appropriate form based on available data
Solve for unknowns systematically
Verify with coordinate geometry principles

📝 Pattern-Based Examples

Example 1: Finding Equation from Points

Question Pattern: Line passing through point and parallel to given line

Problem: Find equation of line through (1,6) parallel to y = 7x + 6

Solution:

G i v e n l in e : y = 7 x + 6 \Rightarrow s l o p e m = 7 P a r a ll e ll in e : s am es l o p e m = 7 P o in t - s l o p e f or m : y - y_{1} = m (x - x_{1}) y - 6 = 7 (x - 1) y - 6 = 7 x - 7 y = 7 x - 1

Answer: y = 7x - 1

Example 2: Intercepts and Systems

Question Pattern: Lines with given intercepts, find intersection

Problem: Line L₁: x-intercept 2, y-intercept -3 Line L₂: x-intercept -1, y-intercept 2 Find intersection point.

Solution:

L₁: x-intercept 2 ⇒ when y=0, x=2 y-intercept -3 ⇒ when x=0, y=-3 Equation: $\dfrac{x}{2} + \dfrac{y}{-3} = 1$ ⇒ 3x - 2y = 6 L₂: x-intercept -1 ⇒ when y=0, x=-1 y-intercept 2 ⇒ when x=0, y=2 Equation: $\dfrac{x}{-1} + \dfrac{y}{2} = 1$ ⇒ -2x + y = -2 Solve system: 3x - 2y = 6 ...(1) -2x + y = -2 ...(2) Multiply (2) by 2: -4x + 2y = -4 Add to (1): -x = 2 ⇒ x = -2 From (2): -2(-2) + y = -2 ⇒ 4 + y = -2 ⇒ y = -6 Intersection: (-2, -6)

⚠️ Common Pitfalls

Mixing up x-intercept and y-intercept
Forgetting negative signs in intercept form
Incorrectly solving systems of equations

📊 Pattern 2: Distance and Shortest Path Problems

📖 Concept Explanation

Distance Formula: Between (x₁,y₁) and (x₂,y₂): $(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$

Distance from Point to Line: ax + by + c = 0: $\frac{∣ a x _{0} + b y _{0} + c ∣}{a ^{2} + b ^{2}}$

Shortest Path: Minimum distance considering constraints

🧠 Mental Algorithm

Convert line to standard form if needed
Apply distance formula correctly
Consider geometric interpretation (perpendicular distance)
Verify units and coordinate system

📝 Pattern-Based Examples

Example 3: Point to Line Distance

Question Pattern: Shortest distance in infrastructure problems

Problem: National highway: line through (2,1) and (10,7) Town locations: A(3,8), B(5,7), C(6,9) Find shortest connecting road.

Solution:

F i rs t, f in d e q u a t i o n o f hi g h w a y : P o in t s (2, 1), (10, 7) Sl o p e m = (7 - 1) / (10 - 2) = 6/8 = 3/4 Eq u a t i o n : y - 1 = (3/4) (x - 2) y - 1 = (3/4) x - 3/2 y = (3/4) x - 1/2 St an d a r df or m : (3/4) x - y - 1/2 = 0 M u lt i pl y b y 4 : 3 x - 4 y - 2 = 0 D i s t an ce f ro m A (3, 8) t o l in e 3 x - 4 y - 2 = 0 : d = ∣3 (3) - 4 (8) - 2∣/\sqrt (9 + 16) = ∣ - 24 - 2∣/5 = 26/5 = 5.2 D i s t an ce f ro m B (5, 7) : d = ∣3 (5) - 4 (7) - 2∣/5 = ∣15 - 28 - 2∣/5 = 15/5 = 3 D i s t an ce f ro m C (6, 9) : d = ∣3 (6) - 4 (9) - 2∣/5 = ∣18 - 36 - 2∣/5 = 20/5 = 4 M inim u m d i s t an ce : 3 u ni t s (P o in tB)

Answer: Location B, distance 300 meters (1 unit = 100m)

Example 4: Collision Detection

Question Pattern: Moving objects, determine intersection

Problem: Bird flying along 2y - 6x = 6, plane from origin through (4,12). Do they collide?

Solution:

B i r d^{'} s p a t h : 2 y - 6 x = 6 \Rightarrow y = 3 x + 3 Pl an e^{'} s p a t h : t h ro ug h (0, 0) an d (4, 12) Sl o p e m = 12/4 = 3 Eq u a t i o n : y = 3 x S e t e q u a l : 3 x + 3 = 3 x \Rightarrow 3 = 0 (co n t r a d i c t i o n) N o in t ersec t i o n \to n oco ll i s i o n

Answer: 0

⚠️ Common Pitfalls

Forgetting absolute value in distance formula
Incorrect line equation conversion
Misinterpreting collision vs parallel paths

🪩 Pattern 3: Reflection Problems

📖 Concept Explanation

Reflection Principle: Angle of incidence = angle of reflection Method: Reflect the endpoint over the mirror line, then connect straight line

🧠 Mental Algorithm

Identify mirror line (usually x-axis or coordinate axes)
Reflect the endpoint over the mirror
Find line equation from start point to reflected endpoint
Find intersection with mirror line

📝 Pattern-Based Examples

Example 5: Light Reflection

Question Pattern: Ray from A reflects at B on mirror, passes through C

Problem: Ray from A(1,2) reflects on x-axis, passes through (5,3). Find equation of AB.

Solution:

R e f l ec tp o in t (5, 3) o v er x - a x i s : (5, - 3) L in e f ro m A (1, 2) t ore f l ec t e d p o in t (5, - 3) : Sl o p e m = (- 3 - 2) / (5 - 1) = (- 5) /4 = - 5/4 Eq u a t i o n : y - 2 = (- 5/4) (x - 1) y - 2 = (- 5/4) x + 5/4 y = (- 5/4) x + 5/4 + 8/4 = (- 5/4) x + 13/4 M u lt i pl y b y 4 : 4 y = - 5 x + 135 x + 4 y - 13 = 0

Answer: 5x + 4y = 13

Example 6: Bird and Plane Collision

Question Pattern: Moving objects with reflection

Problem: Bird: 2y - 6x = 6, plane: slope 2 through (4,8). Find collision point.

Solution:

B i r d : 2 y - 6 x = 6 \Rightarrow y = 3 x + 3 Pl an e : s l o p e 2 t h ro ug h (4, 8) y - 8 = 2 (x - 4) y = 2 x - 8 + 8 = 2 x S e t e q u a l : 3 x + 3 = 2 x \Rightarrow x = - 3 y = 2 (- 3) = - 6 C o ll i s i o n p o in t : (- 3, - 6) S u m : - 3 + (- 6) = - 9

Answer: -9

⚠️ Common Pitfalls

Reflecting wrong point over mirror
Incorrect slope calculation for reflected path
Forgetting to find actual intersection point

🔺 Pattern 4: Area and Triangle Problems

📖 Concept Explanation

Area of Triangle: (1/2)| (x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)) |

Section Formula: Point dividing join of (x₁,y₁) and (x₂,y₂) in m:n

Internal: $(\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n})$
External: $(\frac{m x _{2} - n x _{1}}{m - n}, \frac{m y _{2} - n y _{1}}{m - n})$

Midpoint: $(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$

🧠 Mental Algorithm

List coordinates in order (clockwise/counterclockwise)
Apply area formula carefully with signs
Use section formula for division ratios
Verify with distance checks

📝 Pattern-Based Examples

Example 7: Area with Section Formula

Question Pattern: Triangle with points dividing sides in given ratios

Problem: Triangle ABC with A(-3,3), B(1,7), C(2,-2) M divides AB in 1:3, N divides AC in 2:3, O is midpoint of BC Find area of △MNO.

Solution:

First, find coordinates: M divides AB in 1:3: M = $\left( \dfrac{1\cdot1 + 3\cdot(-3)}{1+3}, \dfrac{1\cdot7 + 3\cdot3}{1+3} \right) = \left( \dfrac{1-9}{4}, \dfrac{7+9}{4} \right) = (-2, 4)$ N divides AC in 2:3: N = $\left( \dfrac{2\cdot2 + 3\cdot(-3)}{2+3}, \dfrac{2\cdot(-2) + 3\cdot3}{2+3} \right) = \left( \dfrac{4-9}{5}, \dfrac{-4+9}{5} \right) = (-1, 1)$ O midpoint BC: O = $\left( \dfrac{1+2}{2}, \dfrac{7+(-2)}{2} \right) = (1.5, 2.5)$ Area of △MNO: M(-2,4), N(-1,1), O(1.5,2.5) Area = (1/2) | -2(1 - 2.5) + (-1)(2.5 - 4) + 1.5(4 - 1) | = (1/2) | -2(-1.5) + (-1)(-1.5) + 1.5(3) | = (1/2) | 3 + 1.5 + 4.5 | = (1/2)(9) = 4.5

Answer: 4.5

Example 8: Parallelogram Vertices

Question Pattern: Finding fourth vertex of parallelogram

Problem: Parallelogram ABCD with A(x₁,y₁), B(x₂,y₂), C(x₃,y₃). Find D.

Solution:

I n p a r a ll e l o g r am, d ia g o na l in t ersec t i o ni s mi d p o in t : M i d p o in t A B = M i d p o in t D CM i d p o in t A D = M i d p o in tBC V ec t or m e t h o d : D = A + (C - B) = (x_{1}, y_{1}) + (x_{3} - x_{2}, y_{3} - y_{2}) = (x_{1} - x_{2} + x_{3}, y_{1} - y_{2} + y_{3}) A lt er na t i v e : D = C + (A - B) = s am eres u lt

Answer: (x₁ - x₂ + x₃, y₁ - y₂ + y₃)

⚠️ Common Pitfalls

Incorrect order of coordinates in area formula
Wrong section formula (internal vs external)
Mixing up parallelogram vertex calculation

💰 Pattern 5: Linear Models & Cost Analysis

📖 Concept Explanation

Cost Functions: Total cost = Fixed cost + Variable cost × quantity Break-even Analysis: Find when revenue = cost Optimization: Minimize cost, maximize profit

🧠 Mental Algorithm

Define variables clearly (cost, quantity, time)
Set up equations for each option
Compare alternatives systematically
Calculate savings or optimal choice

📝 Pattern-Based Examples

Example 9: Mobile Plan Optimization

Question Pattern: Compare service plans with different cost structures

Problem: Mobile cost ₹22,000 + network plan vs separate purchase Network options: Astron (₹100 + ₹2/min), Proton (₹200 + ₹0.5/min) 200 minutes/month needed. Find best option.

Solution:

C o m p an yo ff er : ₹34, 000 (m o bi l e + 1 - ye a r u n l imi t e d) S e p a r a t e p u rc ha se : M o bi l e ₹22, 000 + n e tw or k M o n t h l ycos t co m p a r i so n (200 min u t es) : A s t ro n : 100 + 2 \times 200 = 100 + 400 = ₹500 P ro t o n : 200 + 0.5 \times 200 = 200 + 100 = ₹300 B es t n e tw or k : P ro t o n (₹300/ m o n t h) Y e a r l y n e tw or k cos t : 300 \times 12 = ₹3, 600 T o t a l se p a r a t ecos t : 22, 000 + 3, 600 = ₹25, 600 S a v in g s : 34, 000 - 25, 600 = ₹8, 400

Answer: 8,400

Example 10: Linear Regression

Question Pattern: Best fit line, SSE calculation

Problem: Weight loss data, find if model is good fit (SSE < 5)

Solution Strategy:

Calculate means of x and y
Find best fit line using least squares
Compute SSE = Σ(predicted - actual)²
Compare with threshold

⚠️ Common Pitfalls

Not converting units consistently
Forgetting monthly vs yearly calculations
Incorrect SSE interpretation

📈 Pattern 6: Data Analysis & Curve Fitting

📖 Concept Explanation

Linear Regression: y = mx + c minimizing Σ(yᵢ - (mxᵢ + c))² Sum of Squared Errors (SSE): Σ(predicted - actual)² Goodness of Fit: Compare SSE with threshold

🧠 Mental Algorithm

Understand data pattern (linear, quadratic, etc.)
Calculate required parameters (means, sums)
Apply appropriate model
Evaluate fit quality

📝 Pattern-Based Examples

Example 11: SSE Calculation

Question Pattern: Verify if linear model fits data well

Problem: Monthly expenses vs outings, SSE for y = 4x + 2

Solution:

D a t a : O u t in g s (x) : 1, 3, 5, 7, 9, 11 E x p e n ses (y) : 6, 14, 24, 29, 39, 45 P re d i c t i o n s : x = 1 : 4 (1) + 2 = 6, error = 6 - 6 = 0 x = 3 : 4 (3) + 2 = 14, error = 14 - 14 = 0 x = 5 : 4 (5) + 2 = 22, error = 24 - 22 = 2, s q u a re d = 4 x = 7 : 4 (7) + 2 = 30, error = 29 - 30 = - 1, s q u a re d = 1 x = 9 : 4 (9) + 2 = 38, error = 39 - 38 = 1, s q u a re d = 1 x = 11 : 4 (11) + 2 = 46, error = 45 - 46 = - 1, s q u a re d = 1 SSE = 0 + 0 + 4 + 1 + 1 + 1 = 7

Answer: 7

Example 12: Curve Fitting

Question Pattern: Find parameter to minimize SSE

Problem: f(x) = -(x-1)²(x-3)(x-5)(x-7) + c, find c minimizing SSE

Solution Strategy:

Compute predicted values for each data point
Calculate SSE for different c values
Find minimum SSE

⚠️ Common Pitfalls

Incorrect error calculation (predicted - actual)
Forgetting to square errors
Not understanding SSE threshold interpretation

📚 Pattern-Based Exercises

Set 1: Line Equations (4 questions)

Find equation of line through (2,3) perpendicular to 3x + 4y = 12
Line through (1,1) parallel to line joining (2,3) and (4,7)
Find x-intercept of line through (3,4) with slope -2
Determine if lines 2x + 3y = 6 and 4x + 6y = 12 are parallel

Set 2: Distance Problems (3 questions)

Distance from (0,0) to line 3x + 4y = 12
Shortest distance between parallel lines 2x + 3y = 6 and 2x + 3y = 12
Point closest to origin on line x + 2y = 5

Set 3: Reflection & Collision (3 questions)

Light from (1,2) reflects on y-axis, passes through (3,4)
Two objects moving, find if they intersect
Find intersection point of bird and plane paths

Set 4: Area Problems (3 questions)

Area of triangle with vertices (1,2), (3,4), (5,6)
Find area of quadrilateral with given vertices
Point divides median in 2:1 ratio, find area of small triangle

Set 5: Linear Models (3 questions)

Compare two pricing plans, find cost difference
Break-even analysis for two options
Optimization problem with constraints

🎯 Mental Algorithms Summary

Quick Reference for Exam

Lines: Choose form based on given info (slope, points, intercepts)
Distance: Convert to standard form, apply formula
Reflection: Reflect endpoint, find straight line, intersect with mirror
Area: Use shoelace formula, careful with coordinate order
Optimization: Set up equations, compare alternatives, calculate savings

📈 Progress Tracking

Week 2 Completion Checklist:

Master all 6 question patterns
Complete 15+ practice exercises
Achieve 90%+ accuracy on pattern sets
Review all solution explanations
Identify personal error patterns

Next: Week 3 - Quadratic Functions

Remember: Coordinate geometry is about visualization. Draw sketches, even mentally, to understand the geometric relationships.

🧠 IIT Madras Notes

Explorer

Mathematics 1 - Week 2: Coordinate Geometry & Linear Models

📐 Mathematics 1 - Week 2: Coordinate Geometry & Linear Models

📋 Week 2 Overview

📏 Pattern 1: Lines and Their Equations

📖 Concept Explanation

🧠 Mental Algorithm

📝 Pattern-Based Examples

Example 1: Finding Equation from Points

Example 2: Intercepts and Systems

⚠️ Common Pitfalls

📊 Pattern 2: Distance and Shortest Path Problems

📖 Concept Explanation

🧠 Mental Algorithm

📝 Pattern-Based Examples

Example 3: Point to Line Distance

Example 4: Collision Detection

⚠️ Common Pitfalls

🪩 Pattern 3: Reflection Problems

📖 Concept Explanation

🧠 Mental Algorithm

📝 Pattern-Based Examples

Example 5: Light Reflection

Example 6: Bird and Plane Collision

⚠️ Common Pitfalls

🔺 Pattern 4: Area and Triangle Problems

📖 Concept Explanation

🧠 Mental Algorithm

📝 Pattern-Based Examples

Example 7: Area with Section Formula

Example 8: Parallelogram Vertices

⚠️ Common Pitfalls

💰 Pattern 5: Linear Models & Cost Analysis

📖 Concept Explanation

🧠 Mental Algorithm

📝 Pattern-Based Examples

Example 9: Mobile Plan Optimization

Example 10: Linear Regression

⚠️ Common Pitfalls

📈 Pattern 6: Data Analysis & Curve Fitting

📖 Concept Explanation

🧠 Mental Algorithm

📝 Pattern-Based Examples

Example 11: SSE Calculation

Example 12: Curve Fitting

⚠️ Common Pitfalls

📚 Pattern-Based Exercises

Set 1: Line Equations (4 questions)

Set 2: Distance Problems (3 questions)

Set 3: Reflection & Collision (3 questions)

Set 4: Area Problems (3 questions)

Set 5: Linear Models (3 questions)

🎯 Mental Algorithms Summary

Quick Reference for Exam

📈 Progress Tracking

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