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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.

Note:

  • 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 

Therefore

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)  

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  

\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.

Note:

  • 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.

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  

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