A (straight) line is the simplest of geometrical figures. Imagining a line is very easy. The condition of straightness is also very intuitive. We now formally define a line as a locus of a point satisfying a condition.
NOTE : Throughout this chapter, we will be using the coordinate geometry.
Inclination of a Line
The smallest non-negative angle made by a line with positive axis is known as the inclination. It is generally denoted by .
I) Parallel lines have same inclinations.
II) Inclination of axis is and inclination of axis is .
Slope or Gradient of a Line ,
If is the inclination, then slope is given by .
NOTE : Knowing trigonometric ratios of particular angles is beneficial.
I) Parallel lines have same slope.
II) Slope of axis is and slope of axis is not defined.
III) Product of slopes of perpendicular lines is . This property is extremely important.
If and are 2 distinct points on a line, its slope is given by
Line is locus of points, such that slope of line segment joining any 2 distinct points on the line is same.
NOTE : Since line is a locus of points, it has an equation.
The and intercepts of a line are the and coordinates of the points, where the line meets respective axes. These being directed distances, can be either positive or negative.
Equations of Line in Various Forms
I) Slope- Point Form
Equation of a line passing through point and having a slope is
II) Two-Point Form
Equation of a line passing through points and is given by
III) Slope-Intercept Form
Equation of a line having slope and intercept is given by
IV) Double-Intercept Form
Equation of a line having $X$ and $Y$ intercepts as $a$ and $b$ is given by
V) Normal Form (Rarely Used)
If is the length of perpendicular from origin to the line and is the inclination of the perpendicular segment, then
VI) Parametric Form (More Effective in 3D)
If is the inclination and is the point on a line, then the parametric equations are given by
where is the distance of from
General Equation of 1st Order in 2 Variables
The general equation of 1st order in 2 variables is of the form
It represents a straight line.
The necessary condition is both and cannot be simultaneously zero.
Angle Between Any 2 Lines
Let be the angle between 2 lines having slopes and . Then,
Point of Intersection of 2 Lines
We know that 2 non-parallel lines intersect each other at only 1 point. Let the equations of lines be
The coordinates of point of intersection are
The condition is,
Condition for Concurrency of 3 Lines
Lines are said to be concurrent, when all of them intersect at a point.
Let the lines be , and . The lines are concurrent, when
The point of concurrence can then be obtained easily by considering any 2 of those 3 lines.
Length of Perpendicular from a Point to a Line
Let be a point and be a line. Then, the length of perpendicular from to the line is given by
Distance between 2 Parallel Lines
We know that parallel lines have same slope. This means, in their equations, the ratio of coefficients of and will be same. Let the lines be and . The distance is given by
Family of Straight Lines
This is a useful concept. We know that, through a single point, infinitely many lines pass. Let and be 2 intersecting lines. Then, equation of family of lines passing through the point of intersection is given by
where is a parameter.