Analytical Geometry Formulas

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. ¹⁄₂ {x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)}
Area of quadrilateral	Area of the quadrilateral if four vertices of quadrilateral are given.
	¹⁄₂{(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-y₁)/(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