Basic Concept of a Line

  • Any point, that satisfies the equation of a line, will lie on the line.
  • Also, any point through which a line passes, will always satisfy the equation of that line.
  • Every straight line can be represented by a linear equation in two dimensions ( x \ and \ y plane).

Inclination of a Line

The angle \theta   that the line makes with the x-axis  is called the “inclination of the line”. The angle  \theta  is when measured in anti-clockwise direction the inclination is  positive and if measured clock-wise then the inclination is negative.


  • Inclination angle \theta  for x-axis  is 0^o  . Also all lines parallel to x-axis  will also have the inclination angle as 0^o  .
  • Inclination angle \theta  for y-axis  is 90^o  . Also all lines parallel to y-axis  will also have the inclination angle as 90^o  .

Slope (or Gradient) of a Line

The slope of the line is the tangent of its inclination and is denoted by m  i.e. Slope m = \tan \theta 


Slope  of the x-axis = \tan 0^o= 0 

Slope of the y-axis = \tan 90^o = \infty  (not defined)

3The slope of the line is positive if the line makes an acute angle with x-axis  when measured in anti-clock wise direction  i.e.   \tan \theta  is positive.

The slope of the line is negative if the line makes and obtuse angle with x-axis  when measured in anti-clockwise direction. i.e.   \tan \theta  is negative.

Formula for “Slope of Line passing through two points”

Let the two points be   (x_1, y_1) and   (x_2, y_2)  

\displaystyle m = \tan \theta =  \frac{y_2-y_1}{x_2-x_1} = \frac{y_1-y_2}{x_1-x_2} 

Parallel Lines

1Two lines having inclination as \theta \ and \  \alpha    will be parallel if

\alpha=\theta \ or \  \tan \alpha=tan\theta  

i.e. slope of the two lines is equal and hence they are parallel lines.

Perpendicular Lines

2Two lines having inclination as \theta and \alpha will be perpendicular if:

\theta = 90^o+\alpha  

\Rightarrow \tan \theta=tan \ (90^o+\alpha)  

\Rightarrow \tan \theta=- \cot \alpha  

\displaystyle \Rightarrow \tan \theta=-  \frac{1}{\tan \alpha} 

\Rightarrow \tan \theta \times \tan \alpha = -1  

or m_1 \times m_2 = -1   where m_1 \ and \  m_2   are slopes of the two lines respectively.


  • If two lines are perpendicular, then the product of their slopes is -1 . Conversely, if the product of the two lines is -1 , they are perpendicular.
  • Example: If the slope of a line is 2  , then the slope of another line parallel to this line would be 2  . Also the slope of a line perpendicular to this line would be -  \frac{1}{2}  .
  • A line whose slope is m=0  , is parallel to x-axis  .
  • Also the slope of y-axis    is not defined, all likes parallel to y-axis   will have slope which is not defined.

Condition for Collinearity of Three Points

Let there be three points A(x_1, y_1), B(x_2, y_2) \  and \ C(x_3, y_3)

If the slope of segment AB and BC is the same, then they are all on the same line and are collinear.

\displaystyle m  =  \frac{y_2-y_1}{x_2-x_1} = \frac{y_3-y_2}{x_3-x_2}

X-intercepts & Y-intercepts

Refer to first figure.

If a line meets the x-axis , then the distance from origin to the point of intercept is called x-intercept .

Similarly, if a line meets the  y-axis , then the distance from origin to the point of intercept is called  y-intercept  .

Equation of a Line

Slope-Intercept Form: When the slope (m)  is given and the intercept on the y-axis  is given.

The the equation is:  y = mx+c  where m  is the slope and c = intercept \ on \  y-axis .

Point-Slope form: When the slope of the line and a point (x_1, y_1)  on a line is given

The equation is:  y-y_1 = m(x-x_1)  

Equally Inclined Lines

This means that the lines make equal angle from both the coordinate axes. As shown in the figure, lines AB \ and \ CD   are equally inclined.

For AB: Inclination \  \theta=45^o  . Therefore slope = \tan 45^o = 1  

For CD: Inclination \  \theta=-45^o  . Therefore the slope = \tan (-45^o) = -1