# Line (in 2 Dimensions)

• ### Definition

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 ${X}$ axis is known as the inclination.  It is generally denoted by ${\theta}$.

I) Parallel lines have same inclinations.

II) Inclination of ${X}$ axis is ${0^o}$ and inclination of ${Y}$ axis is ${90^o}$.

• ### Slope or Gradient of a Line , ${m}$

If ${\theta}$ is the inclination, then slope is given by ${tan (\theta)}$.

NOTE : Knowing trigonometric ratios of particular angles is beneficial.

I) Parallel lines have same slope.

II) Slope of ${X}$ axis is ${0}$ and slope of ${Y}$ axis is not defined.

III) Product of slopes of perpendicular lines is ${-1}$. This property is extremely important.

If ${(x_1, y_1)}$ and ${(x_2, y_2)}$ are 2 distinct points on a line, its slope is given by

${m = \frac {y_2 - y_1}{x_2 - x_1}}$

• ### Formal Definition

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.

• ### Intercepts

The ${X}$ and ${Y}$ intercepts of a line are the ${X}$ and ${Y}$ 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 ${(x_1,y_1)}$ and having a slope ${m}$ is

${y- y_1 = m (x-x_1)}$

II) Two-Point Form

Equation of a line passing through points ${(x_1,y_1)}$ and ${(x_2,y_2)}$ is given by

${\frac {y-y_1}{y_2 - y_1} = \frac {x - x_1}{x_2 - x_1}}$

III) Slope-Intercept Form

Equation of a line having slope ${m}$ and ${y}$ intercept ${c}$ is given by

${y = mx +c}$

IV) Double-Intercept Form

Equation of a line having $X$ and $Y$ intercepts as $a$ and $b$ is given by

${\frac {x}{a} + \frac {y}{b} = 1}$

V) Normal Form (Rarely Used)

If ${p}$ is the length of perpendicular from origin to the line and ${\alpha}$ is the inclination of the perpendicular segment, then

${x \ cos (\alpha) + y \ sin (\alpha) = p}$

VI) Parametric Form (More Effective in 3D)

If ${\theta}$ is the inclination and ${(x_1,y_1)}$ is the point on a line, then the parametric equations are given by

${x = x_1 + r \ cos (\theta) \ and \ y = y_1 + r \ sin (\theta),}$

where ${r}$ is the distance of ${(x,y)}$ from ${(x_1, y_1)}$

• ### General Equation of 1st Order in 2 Variables

The general equation of 1st order in 2 variables is of the form

${ax + by + c = 0}$

It represents a straight line.

The necessary condition is both ${a}$ and ${b}$ cannot be simultaneously zero.

• ### Angle Between Any 2 Lines

Let ${\theta}$ be the angle between 2 lines having slopes ${m_1}$ and ${m_2}$. Then,

${tan (\theta) = \begin {vmatrix} \frac {m_1 - m_2}{1+ m_1 m_2} \end {vmatrix}}$

• ### 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

${a_1 x + b_1 y = c_1 \ and \ a_2 x + b_2 y = c_2}$

The coordinates of point of intersection are

${\begin {pmatrix}\frac {\begin {vmatrix} c_1 & c_2 \\ b_1 & b_2 \end {vmatrix}}{\begin {vmatrix} a_1 & a_2 \\ b_1 & b_2 \end {vmatrix}} , \frac {\begin {vmatrix} a_1 & a_2 \\ c_1 & c_2 \end {vmatrix}}{\begin {vmatrix} a_1 & a_2 \\ b_1 & b_2 \end {vmatrix}} \end {pmatrix}}$

The condition is,

${\begin {vmatrix} a_1 & a_2 \\ b_1 & b_2 \end {vmatrix} \ne 0}$

• ### Condition for Concurrency of 3 Lines

Lines are said to be concurrent, when all of them intersect at a point.

Let the lines be ${a_1 x + b_1 y + c_1 = 0}$ , ${a_2x + b_2 y + c_2 = 0}$ and ${a_3x + b_3 y + c_3 = 0}$ . The lines are concurrent, when

${\begin {vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end {vmatrix} = 0}$

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 ${P (x_1,y_1)}$ be a point and ${ax+by+c=0}$ be a line. Then, the length of perpendicular from ${P}$ to the line is given by

${\begin {vmatrix} \frac {ax_1 + by_1 + c}{\sqrt {a^2+b^2}} \end {vmatrix}}$

• ### Distance between 2 Parallel Lines

We know that parallel lines have same slope. This means, in their equations, the ratio of coefficients of ${x}$ and ${y}$ will be same. Let the lines be ${ax+by+c_1 = 0}$ and ${ax+by+c_2 = 0}$. The distance is given by

${\begin {vmatrix} \frac {c_1 - c_2}{\sqrt {a^2+b^2}} \end {vmatrix}}$

• ### Family of Straight Lines

This is a useful concept. We know that, through a single point, infinitely many lines pass. Let ${u \equiv a_1x + b_1 y + c_1 =0}$ and ${v \equiv a_2 x + b_2 y + c_2 =0}$ be 2 intersecting lines. Then, equation of family of lines passing through the point of intersection is given by

${u + \lambda v = 0, \ \lambda \in \mathbb R,}$

where ${\lambda}$ is a parameter.