01. Cartesian Plane
The Cartesian plane is formed by two perpendicular number
lines:
• Horizontal line → x-axis
• Vertical line → y-axis
• Their intersection → Origin O(0,0)
💡 Easy Hindi:
Coordinate plane ko ek map samjho. Kisi bhi point ki exact position
batane ke liye hum uske coordinates `(x,y)` use karte hain.
Pehle x-coordinate, phir y-coordinate likha jata hai.
02. Coordinates of a Point
A point is represented as:
P(x,y)
Here:
x → abscissa
y → ordinate
✏ Example
Find the abscissa and ordinate of P(−4, 7).
Abscissa = x-coordinate =
−4
Ordinate = y-coordinate =
7
✓ Answer: Abscissa = −4, Ordinate = 7
03. Signs in Four Quadrants
| Quadrant |
x |
y |
| I |
+ |
+ |
| II |
− |
+ |
| III |
− |
− |
| IV |
+ |
− |
✏ Example
Point A(−3, 5) lies in which quadrant?
x is negative and y is positive.
✓ A lies in Quadrant II.
04. Distance Between Two Points ⭐⭐⭐
For two points:
A(x₁,y₁) and B(x₂,y₂)
the distance AB is given by the Distance Formula.
💡 Hindi:
Do points ke beech straight-line distance nikalne ke liye
x ka difference aur y ka difference lekar Pythagoras theorem
apply karte hain.
✏ Example 1 — Direct Formula
Find the distance between A(2,3) and B(6,6).
x₁ = 2, y₁ = 3
x₂ = 6, y₂ = 6
AB = √[(6−2)² + (6−3)²]
= √[4² + 3²]
= √25
= 5 units
✓ Distance = 5 units
05. Distance from Origin ⭐⭐
✏ Example
Find the distance of P(3,4) from origin.
OP = √(3² + 4²)
= √25
= 5 units
06. Points on Same Vertical / Horizontal Line
Same x-coordinate
A(x,y₁), B(x,y₂)
Distance:
|y₂ − y₁|
Same y-coordinate
A(x₁,y), B(x₂,y)
Distance:
|x₂ − x₁|
✏ Example
Find the distance between A(5,2) and B(5,9).
Both have the same x-coordinate.
Distance = |9−2| =
7 units
✓ Answer = 7 units
07. Finding an Unknown Coordinate Using Distance
✏ Exam-Type Example
Find x if the distance between A(x,2) and B(3,6) is 5 units.
5 = √[(3−x)² + (6−2)²]
25 = (3−x)² + 16
(3−x)² = 9
3−x = ±3
Therefore:
x = 0 or x = 6
💡 Equation ko solve karte time square root remove karne ke baad
± possibility ko ignore mat karo.
08. Section Formula ⭐⭐⭐
Suppose a point P divides the line segment joining
A(x₁,y₁) and B(x₂,y₂)
internally in the ratio:
AP : PB = m : n
Then the coordinates of P are given by the Section Formula.
💡 Hindi trick:
Jis point ko find karna hai, uske coordinates mein opposite
ratio ka multiplication aata hai.
For P dividing AB in m:n:
x-coordinate = `(m × x₂ + n × x₁)/(m+n)`
y-coordinate = `(m × y₂ + n × y₁)/(m+n)`
09. Section Formula — Direct Example ⭐⭐⭐
✏ Example
Find the coordinates of P which divides A(2,3) and B(8,9)
internally in the ratio 1:2.
10. Midpoint Formula ⭐⭐⭐
Midpoint is the point which divides a line segment into
two equal parts.
So:
AP : PB = 1 : 1
Therefore midpoint formula is obtained from section formula.
✏ Example
Find the midpoint of A(2,4) and B(8,10).
M =
(
(2+8)/2 ,
(4+10)/2
)
= (5,7)
✓ Midpoint = (5,7)
11. Finding a Missing Coordinate Using Midpoint
✏ Exam-Type Example
The midpoint of A(x,4) and B(6,8) is (4,6).
Find x.
Midpoint x-coordinate:
4 = (x + 6)/2
8 = x + 6
x = 2
Check y:
(4+8)/2 = 6 ✓
✓ A = (2,4)
12. Finding Ratio When Point is Given
✏ Exam-Type Example
Point P(4,5) divides A(2,3) and B(8,9).
Find AP:PB.
13. Internal Division — Figure Based Question
✏ Example
A point P divides the line joining A(1,2) and B(7,8)
internally in the ratio 2:1.
Find P.
14. Proving a Triangle is Isosceles
Coordinate geometry can be used to prove geometric properties
by calculating lengths.
For an isosceles triangle:
Two sides must be equal.
✏ Example
Show that A(0,0), B(4,0), C(2,3) form an isosceles triangle.
15. Proving a Triangle is Right-Angled
✏ Exam-Type Example
Check whether A(0,0), B(3,0), C(3,4) form a right-angled triangle.
💡 Teen points se triangle right-angled prove karne ke liye
sabse badi distance ko hypotenuse maan kar Pythagoras check karo.
16. Important Exam Pattern — Equidistant Point
✏ Example
Find the point on the x-axis which is equidistant from
A(2,3) and B(4,5).
17. Important Formula Sheet