ANALYTICAL GEOMETRY FORMULAS

Name of the topic

Formula

Section formula  (Internally)

The formula to find the point which is dividing the line segment AB internally in the ratio m:n is given by

Section formula  (externally)

The formula which is used to find the point which divides the line segment AB externally in the ratio m:n is given by

Distance between two points

To find the distance between two points A and B

d = √(x₂ - x₁) ² + (y₂ - y₁) ²

Area of triangle using three vertices

Area of triangle if three vertices of triangle are given.

12 {x1(y2-y3) + x2(y3-y1) + x3(y1-y2)}

Area of quadrilateral

Area of the quadrilateral if four vertices of quadrilateral are given.

12{(x1y2+x2y3+x3y4+x4y1)-
(x2y1+x3y2+x4y3+x1y4)}

Centroid of the triangle

There are three medians of the triangle and they are concurrent at a point O,that point is called the centroid of a triangle.

In the following diagram O is the centroid of ABC.Now let us look into the formula.

       =  (x1+x2+x3)/3, (y1+y2+y3)/3

Midpoint of the line segment

Midpoint is the point which is exactly in the middle of the line segment joining two points (x1,y1) and (x2,y2)

(x₁ + x₂)/2 , (y₁ + y₂)/2

Slope of the line

The angle theta between the straight line and the positive direction of the X axis when measured in the anticlockwise direction is called angle of inclination.The tangent of the angle of inclination is called slope or gradient of the line.

m = tan θ

m = (y2 - y1)/(x2 - x1)

m = - coefficient of x /coefficient of y

y = m x + b

m-slope

Equation of the line

A linear equation or an equation of the first degree in x and y represents a straight line.The equation of a straight line is satisfied by the co-ordinates of every point lying on the straight line and not by any other point outside the straight line.

Slope intercept form:

y = m x + b

Here m = slope and b = y-intercept

Two point form:

(y-y)/(y₂-y₁) = (x-x₁)/(x₂-x₁)

Point- Slope form:

(y-y1) = m (x-x1)

Intercept form:

(X/a) + (Y/b) = 1

Perpendicular distance a point and a line

The length of the perpendicular from the point (x₁,y₁) to the line ax + by + c = 0 is

     d =   | (ax₁ + by₁ + c)/ va² + b² |

Distance between two parallel lines

Distance between two parallel lines

a x + b y + c₁ = 0 and a x + b y + c₂ = 0

    d =   | (c₁ - c₂)/ va² + b² |

Angle between two lines

θ = tan-¹ |(m₁ - m₂)/(1 + m₁ m₂)|

Equation of circle

(x-h)² + (y-k)² = r²

Equation of circle with two endpoints of diameter

(x-x₁) (x-x₂) + (y-y₁) (y-y₂) = 0

General equation of circle

x² + y² + 2gx + 2fy + c = 0

Length of the tangent

√ (x₁² + y₁² + 2gx₁ +2fy₁+c)

Condition of two circles touching externally

c₁c₂ = r₁ + r₂

Condition of two circles touching internally

 C₁ C₂ = r₁ - r₂

Orthogonal circles

2 g₁g₂+2f₁f₂=c₁+c₂


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