Collinearity of three points
2.3.12· Maths › Coordinate Geometry
Core Concept
WHY? Ek line 1-dimensional hoti hai—uski length hoti hai lekin width nahi. Agar teen points collinear hain, toh woh koi bhi 2D region enclose nahi kar sakte, isliye jo "triangle" woh banate hain woh degenerate hota hai (collapsed).
The Area Method: Deriving from First Principles
WHAT we're doing: Triangle ka area formula use karke collinearity detect karna.
Starting point: Ek triangle ka area jiske vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) hain:
WHY this formula? Yeh determinant method (2D mein cross product) se aata hai. Agar hum A se B aur A se C tak vectors rakhein, unka cross product area ka do guna deta hai.
Derivation:
- Vector AB = (x₂ - x₁, y₂ - y₁)
- Vector AC = (x₃ - x₁, y₃ - y₁)
- Area = ½|AB × AC| = ½|(x₂ - x₁)(y₃ - y₁) - (x₃ - x₁)(y₂ - y₁)|
- Expanding: ½|x₂y₃ - x₂y₁ - x₁y₃ + x₁y₁ - x₃y₂ + x₃y₁ + x₁y₂ - x₁y₁|
- Regrouping: ½|x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
WHY absolute value disappear ho jaata hai? Collinearity ke liye humein Area = 0 chahiye, isliye andar ka expression zero ke barabar hona chahiye. Zero ke liye absolute value irrelevant hai.
The Slope Method: Alternative Approach
WHAT: Agar teen points collinear hain, toh kisi bhi do pairs ke beech ka slope barabar hona chahiye.
WHY? Ek line ka slope har jagah constant hota hai. Agar slope(AB) = slope(BC), toh B koi direction change nahi karta.
Cross-multiplying (division by zero se bachne ke liye): $(y_2 - y_1)(x_3 - x_2) = (y_3 - y_2)(x_2 - x_1)$$
\text{Area} = \frac{1}{2} |x_A(y_B - y_C) + x_B(y_C - y_A) + x_C(y_A - y_B)|
WHY this step? Hum coordinates directly formula mein substitute karte hain. Result: Haan, colinear hai! (Expression zero ke barabar hai.)
Verification (Slope Method):
- Slope AB = (4-2)/(3-1) = 2/2 = 1
- Slope BC = (6-4)/(5-3) = 2/2 = 1
- Barabar slopes → collinear ✓
Solution:
WHY this matters? Non-zero result ka matlab hai ki triangle ki area = ½|-2| = 1 hai, isliye points ek real triangle banate hain. Result: NOT collinear.
Solution:
WHY this step? Hum k ko unknown maante hain aur colinearity equation solve karte hain. Verification: Slope AB = (5-3)/(4-2) = 1, Slope BC = (7-5)/(6-4) = 1 ✓
## Common Mistakes
Why it feels right: Area formula mein absolute value hoti hai, isliye students usse include kar lete hain.
The fix: Collinearity check karne ke liye, humein expression zero ke barabar chahiye. |kuch bhi| = 0 ka matlab hai andar wala zero hai, isliye hum absolute value turant drop kar sakte hain. Use include karna math ko break nahi karta lekin redundant hai.
Why it feels right: Cyclic pattern similar lagta hai, mix up karna aasaan hai.
The fix: Formula cyclic lekin specific hai: x₁(y₂ - y₃), x₂(y₃ - y₁), x₃(y₁ - y₂). Har x-coordinate baaki DO y-coordinates ka difference multiply karta hai. Mnemonic: "1 khud ko skip karta hai, 2 aur 3 ko multiply karta hai."
Why it feels right: Slope method simple hai, students pehle wohi use karte hain.
The fix: Agar koi bhi do x-coordinates barabar hain, check karo ki TEENO x-coordinates barabar hain ya nahi (vertical line case). Warna, area method use karo jo hamesha kaam karta hai.
Active Recall Practice
#flashcards/maths
Teen points ke collinear hone ki condition kya hai? :: Teen points A(x₁,y₁), B(x₂,y₂), C(x₃,y₃) collinear hain agar x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂) = 0
Colinearity ka matlab zero area kyun hota hai?
Collinearity ke liye slope condition kya hai?
Slope method kab fail ho jaata hai?
Agar points A(1,2), B(3,k), C(5,6) collinear hain, toh k kya hai?
Non-zero area value tumhe kya batata hai?
Feynman Technique
Recall Ek 12 saal ke bachche ko samjhao
Imagine karo tum apne ghar (point A) se apne dost ke ghar (point B) se park (point C) tak chal rahe ho. Agar teeno jagahein ek hi sadak par hain aur tumhe kabhi bhi mudna nahi padta, toh woh collinear hain—ek hi line par!
Ab, agar tumhe park jaane ke liye apne dost ke ghar par MUDNA padta hai, toh teeno jagahein ek triangle banati hain. Woh triangle map par jagah gherta hai.
Yeh trick hai: mathematicians measure karte hain ki woh triangle kitni jagah (area) leta hai. Agar area ZERO hai, iska matlab hai tumne kabhi nahi muda—teeno points ek straight line hain!
Hum coordinates (jaise map par addresses) ke saath ek magic formula use karte hain. Numbers plug in karo, math karo, aur agar zero aaye, boom—woh collinear hain! Agar koi aur number aaye, toh woh ek hi line par nahi hain.
Memory Aid
Har x-coordinate "khud ko skip karta hai" aur baaki do y-coordinates ka difference multiply karta hai. Pattern cyclic hai: (2-3), (3-1), (1-2).
Connections
- Area of Triangle using Coordinates - Woh parent formula jisse colinearity derive hoti hai
- Slope of a Line - Collinearity check karne ka alternative method
- Section Formula - Ek line segment par points nikalne ke liye use hota hai, collinearity se related
- Determinants - Area formula actually ek 3×3 determinant hai
- Vectors and Cross Product - 2D mein geometric interpretation
- Parametric Equations - Colinear points express karne ka ek aur tarika: sabhi r = a + t(b-a) par hote hain
- Linear Dependence - Linear algebra mein, collinear vectors linearly dependent hote hain
Study Tip: Calculate karne se PEHLE random points plot karke andaza lagao ki woh collinear hain ya nahi. Algebraic skill ke saath geometric intuition bhi develop karo. 80/20 yeh hai: Area method master karo (100% cases handle karta hai), quick mental checks ke liye slope method use karo.