Coordinate Geometry | SuccessSetu

SuccessSetu

📘 Quadrants

Plane is divided into four quadrants.

I (+,+) | II (-,+) | III (-,-) | IV (+,-)

Origin O = (0,0)

📘 Coordinates of Point

Any point is represented as:

P = (x , y)

x → Abscissa, y → Ordinate

📘 Equation of Axes

X-axis → y = 0
Y-axis → x = 0

📘 Polar Coordinates

Point represented using distance & angle.

P = (r cosθ , r sinθ)

📘 Mid Point Formula

M = ( (x₁+x₂)/2 , (y₁+y₂)/2 )

📘 Distance Formula

√[(x₂-x₁)² + (y₂-y₁)²]

📘 Section Formula (Internal)

P = ( (m₁x₂+m₂x₁)/(m₁+m₂) , (m₁y₂+m₂y₁)/(m₁+m₂) )

📘 Trigonometric Form

sinθ = y/r
cosθ = x/r
tanθ = y/x

✍️ Practice 1

Find midpoint of (2,4) & (6,8)

Use midpoint formula

✍️ Practice 2

Find distance between (1,2) & (4,6)

Apply distance formula

📘 External Division

External division of line segment.

P = ( (m₁x₂ − m₂x₁)/(m₁ − m₂) , (m₁y₂ − m₂y₁)/(m₁ − m₂) )

📘 Slope of Line

Slope measures steepness of line.

m = tanθ = (y₂ − y₁)/(x₂ − x₁)

📘 Standard Equation

ax + by + c = 0

Slope = −a/b

📘 Point Form

y − y₁ = m(x − x₁)

📘 Slope Intercept Form

y = mx + c

📘 Parallel Lines

If two lines are parallel:

m₁ = m₂

📘 Perpendicular Lines

If two lines are perpendicular:

m₁ × m₂ = −1

📘 Compare Slopes

m = −a/b

From ax + by + c = 0

📘 Angle Between Lines

tanθ = |(m₁ − m₂)/(1 + m₁m₂)|

✍️ Practice 1

Find slope of line joining (2,3) & (6,7).

Use slope formula

✍️ Practice 2

Check if lines are parallel.

Compare slopes

📘 Intercept Form

Intercept at x-axis = a
Intercept at y-axis = b

x/a + y/b = 1

Line cuts x-axis at (a,0) and y-axis at (0,b)

📘 Parallel Lines

Condition for parallel lines:

m₁ = m₂

📘 Perpendicular Lines

Condition for perpendicular lines:

m₁ × m₂ = −1

📘 Coincident Lines

Infinite solutions when:

a₁/a₂ = b₁/b₂ = c₁/c₂

📘 Concurrent Lines

Lines passing through one point.

All lines meet at same point

📘 Distance Between Axes

Intercept length:

√(a² + b²)

📘 Angle Between Two Lines

tanθ = |(m₁ − m₂)/(1 + m₁m₂)|

📘 Distance From Point

Distance of point from line:

|ax₁ + by₁ + c| / √(a² + b²)

✍️ Practice 1

Find intercept form of line: 2x + 3y = 6

Convert into x/a + y/b = 1

✍️ Practice 2

Check if 2x+3y=4 and 4x+6y=8 are parallel.

Compare slopes

📘 Distance Between Parallel Lines

For lines:

ax + by + c₁ = 0
ax + by + c₂ = 0

Distance = |c₁ − c₂| / √(a² + b²)

📘 Reflection (Origin)

Reflection about origin:

(x, y) → (−x, −y)

📘 Reflection on X-Axis

(x, y) → (x, −y)

📘 Reflection on Y-Axis

(x, y) → (−x, y)

📘 Reflection on y = x

(x, y) → (y, x)

📘 Reflection on y = −x

(x, y) → (−y, −x)

📘 Centroid

G = (x₁+x₂+x₃)/3 , (y₁+y₂+y₃)/3

📘 Incentre

I = (ax₁+bx₂+cx₃)/(a+b+c) , (ay₁+by₂+cy₃)/(a+b+c)

📘 Equation of Circle

(x − a)² + (y − b)² = r²

📘 Circle at Origin

x² + y² = r²

📘 Area of |x| + |y| = a

Area = 2a²

✍️ Practice

Find image of (3, −5) in y = −x

Swap + change sign

📘 Equation of Y-Axis

x = 0

Y-axis par har point ka x-coordinate zero hota hai.

📘 Line Parallel to Y-Axis

x = ± a

Y-axis ke parallel sabhi lines vertical hoti hain.

📘 Graph Representation

x = a → Right side line
x = −a → Left side line

📘 Point of Intersection

For two lines:

a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0

x = (b₁c₂ − b₂c₁)/(a₁b₂ − a₂b₁)
y = (a₂c₁ − a₁c₂)/(a₁b₂ − a₂b₁)

✍️ Practice

Find intersection of:

2x + y − 5 = 0
x − y + 1 = 0

📘 Area of Triangle

For vertices (x₁,y₁),(x₂,y₂),(x₃,y₃)

Area = ½ | x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂) |

📘 Special Case

If area = 0 → Points are collinear

📘 Important Tip

Always take absolute value (|)

📘 Triangle on Axis

Area = ½ × Base × Height

✍️ Practice

Find area of:

A(1,2), B(4,6), C(6,3)