Coordinate Geometry: Locus and Section Formula
The Treasure Map Adventure 🗺️
Imagine you’re a pirate with a magical treasure map. But this isn’t an ordinary map—it shows you paths where treasure might be hidden, not just single spots. That’s exactly what locus means in math!
What is Locus?
Locus is a fancy word for a path or collection of points that follow a specific rule.
Think of it Like This:
- A dog tied to a pole with a 5-meter rope
- The dog can walk in a circle around the pole
- That circle is the locus of all positions the dog can reach!
Simple Definition: Locus = All points that satisfy a given condition
Real-Life Locus Examples
| Condition | Locus |
|---|---|
| Same distance from a point | Circle |
| Same distance from two points | Straight line (perpendicular bisector) |
| Same distance from a line | Parallel line |
| Sum of distances from two points is constant | Ellipse |
Locus Examples Around You
🎡 The Ferris Wheel
Every seat on a Ferris wheel follows a circular locus. Why? Each seat stays the same distance from the center!
🚗 Car Turning
When a car turns, the inside wheels trace a smaller circle than the outside wheels—two different circular loci!
✈️ Airplane Path
A plane flying exactly 10 km above ground follows a locus that’s a curved surface parallel to Earth.
Example: A point moving so it's always
3 units from origin (0,0)
The locus is a CIRCLE with:
- Center: (0, 0)
- Radius: 3 units
Locus in the Coordinate Plane
Now let’s put our locus on a coordinate grid—like putting our treasure map on graph paper!
The Magic Formula Approach
When a point P(x, y) moves following a rule, we can write that rule as an equation.
graph TD A["Start: Define the condition"] --> B["Let point be P x,y"] B --> C["Write condition using x and y"] C --> D["Simplify the equation"] D --> E["Final: Equation of Locus"]
Example: Dog on a Leash
Condition: Point P is always 5 units from origin O(0,0)
Step 1: Let P = (x, y)
Step 2: Distance from O to P = 5
Step 3: Using distance formula:
√(x² + y²) = 5
Step 4: Square both sides:
x² + y² = 25
This is the equation of locus—a circle!
Equation of Locus
The equation of locus is simply the mathematical rule written as an equation.
Recipe to Find Equation of Locus:
- Take a general point P(x, y) on the locus
- Write the condition in terms of x and y
- Simplify to get a neat equation
- That’s your locus equation!
Example: Equidistant from Two Points
Problem: Find the locus of points equidistant from A(2, 0) and B(0, 4)
Solution:
Let P(x, y) be any point on the locus
PA = PB (given condition)
√[(x-2)² + (y-0)²] = √[(x-0)² + (y-4)²]
Squaring both sides:
(x-2)² + y² = x² + (y-4)²
x² - 4x + 4 + y² = x² + y² - 8y + 16
-4x + 4 = -8y + 16
-4x + 8y = 12
Equation of locus: x - 2y + 3 = 0
This is a straight line—the perpendicular bisector of AB!
Internal Division Formula
Imagine you and your friend are sharing a chocolate bar. If you divide it in the ratio 2:3, you get the smaller piece and your friend gets the larger one!
The Formula
If point P divides line segment joining A(x₁, y₁) and B(x₂, y₂) internally in ratio m:n, then:
m·x₂ + n·x₁ m·y₂ + n·y₁
P(x,y) = ───────────── , ─────────────
m + n m + n
Memory Trick 🧠
“Cross multiply and add, then divide by sum”
- Multiply x₂ by m, x₁ by n, add them
- Divide by (m + n)
Example: Finding the Point
Problem: Find point P that divides A(1, 2) and B(7, 8) in ratio 2:1 internally.
2(7) + 1(1) 14 + 1 15
x = ───────────── = ──────── = ──── = 5
2 + 1 3 3
2(8) + 1(2) 16 + 2 18
y = ───────────── = ──────── = ──── = 6
2 + 1 3 3
Point P = (5, 6)
External Division Formula
What if instead of dividing a chocolate bar between you, your friend extends it and marks a point outside? That’s external division!
The Formula
If point P divides line segment joining A(x₁, y₁) and B(x₂, y₂) externally in ratio m:n, then:
m·x₂ - n·x₁ m·y₂ - n·y₁
P(x,y) = ───────────── , ─────────────
m - n m - n
Key Difference from Internal Division:
- Internal: Add (m + n)
- External: Subtract (m - n)
Example: External Point
Problem: Find point P that divides A(2, 3) and B(4, 7) in ratio 3:2 externally.
3(4) - 2(2) 12 - 4 8
x = ───────────── = ──────── = ──── = 8
3 - 2 1 1
3(7) - 2(3) 21 - 6 15
y = ───────────── = ──────── = ──── = 15
3 - 2 1 1
Point P = (8, 15)
Notice: P is outside segment AB!
Centroid Using Coordinates
What’s a Centroid?
The centroid is the balancing point of a triangle. If you cut a triangle from cardboard and put a pencil tip at the centroid, it would balance perfectly!
graph TD A["Triangle ABC"] --> B["Draw all 3 medians"] B --> C["Medians meet at ONE point"] C --> D["That point = CENTROID G"]
The Centroid Formula
For triangle with vertices A(x₁, y₁), B(x₂, y₂), C(x₃, y₃):
x₁ + x₂ + x₃ y₁ + y₂ + y₃
Centroid G = ───────────── , ─────────────
3 3
Super Simple: Just AVERAGE the coordinates!
Example: Finding the Centroid
Problem: Find centroid of triangle with A(0, 0), B(6, 0), C(3, 9).
0 + 6 + 3 9
x = ─────────── = ──── = 3
3 3
0 + 0 + 9 9
y = ─────────── = ──── = 3
3 3
Centroid G = (3, 3)
Fun Fact! 🎯
The centroid divides each median in the ratio 2:1 from vertex to midpoint!
Incenter Coordinates
What’s an Incenter?
The incenter is where all three angle bisectors of a triangle meet. It’s also the center of the largest circle that fits inside the triangle!
graph TD A["Triangle ABC"] --> B["Draw angle bisector from A"] A --> C["Draw angle bisector from B"] A --> D["Draw angle bisector from C"] B --> E["All meet at INCENTER I"] C --> E D --> E E --> F["Incircle touches all 3 sides"]
The Incenter Formula
For triangle with vertices A(x₁, y₁), B(x₂, y₂), C(x₃, y₃) and opposite sides a, b, c:
a·x₁ + b·x₂ + c·x₃ a·y₁ + b·y₂ + c·y₃
Incenter = ─────────────────── , ───────────────────
a + b + c a + b + c
Where:
- a = length of side opposite to vertex A (i.e., BC)
- b = length of side opposite to vertex B (i.e., CA)
- c = length of side opposite to vertex C (i.e., AB)
Memory Trick 🧠
“Weight each vertex by its OPPOSITE side”
Example: Finding the Incenter
Problem: Find incenter of triangle with A(0, 0), B(4, 0), C(0, 3).
Step 1: Find side lengths
a = BC = √[(4-0)² + (0-3)²] = √(16+9) = 5
b = CA = √[(0-0)² + (0-3)²] = 3
c = AB = √[(4-0)² + (0-0)²] = 4
Step 2: Apply formula
5(0) + 3(4) + 4(0) 0 + 12 + 0 12
x = ───────────────────── = ────────── = ──── = 1
5 + 3 + 4 12 12
5(0) + 3(0) + 4(3) 0 + 0 + 12 12
y = ───────────────────── = ────────── = ──── = 1
5 + 3 + 4 12 12
Incenter I = (1, 1)
Quick Summary Table
| Concept | Formula | Remember |
|---|---|---|
| Locus | Path following a condition | Dog on leash! |
| Equation of Locus | Simplify condition with P(x,y) | Write, simplify, done! |
| Internal Division | (mx₂+nx₁)/(m+n) | Add & divide |
| External Division | (mx₂-nx₁)/(m-n) | Subtract & divide |
| Centroid | Average of coordinates | Balance point |
| Incenter | Weighted by opposite sides | Angle bisectors meet |
You Did It! 🎉
You’ve just learned:
- ✅ What locus means and how to spot it
- ✅ How to find the equation of locus
- ✅ Internal and external division formulas
- ✅ Finding the centroid (balance point)
- ✅ Finding the incenter (inscribed circle center)
Remember: These formulas are like GPS coordinates—they help you find exact locations in the coordinate plane!
Now go forth and divide lines, find centers, and trace beautiful loci! 🚀