Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Inclination of a line

Inclination of a line is a real number  \theta as defined below:

(i) If a line   L is parallel to  x– axis or coincides with x -axis then  \theta =0

 L \parallel x- \text {axis}

(ii) If a line is not parallel to  x-axis  then let it intersect  x -axis at a point  P .

Inclination  \theta of  L is given by   \theta =m < APx

(iii) Inclination of  \overline{AB},\overrightarrow{AB}   or  \overrightarrow{BA}   is defined as the inclination of \overleftrightarrow{AB}


 \to If \theta    is the inclination of a line then  0 \leq \theta < \pi .

 \to Parallel lines have same inclination and conversely.

 \to Inclination of perpendicular line differ by  \dfrac{\pi}{2}.

 \to Inclination is essentially an angle-measure. The only difference is that inclination can be zero, where as angle-measure can not be zero.

Slope (Gradient) of Non-Vertical Line:

The slope of a non-vertical line is given by m=\tan \theta , where  \theta   is the inclination of the line.


\to Inclination \theta \in [0,\pi] ,  but slope  \in R.

\to Slope of a vertical line is not defined.

\to Slope of a line is positive, zero or negative according as the inclination of the line is less than  \dfrac{\pi}{2} , equal to  \dfrac{\pi}{2} or greater than  \dfrac{\pi}{2} respectively.

\to It is obvious that slope of a line is the ration of its rise or fall  (y_2-y_1) to its run (x_2-x_1)

A line with positive slope looks like rising where a line with negative slope looks like falling as we move from left to right along the line.

\text { Slope} (m) =\dfrac{y_2-y_1}{x_2-x_1}

\to \dfrac{y_2-y_1}{x_2-x_1}=\dfrac{\triangle y}{\triangle x}

Positive Slope:


Rise= Positive ; Run =Positive

Negative Slope:

Rise= Negative ; Run = Positive

Zero Slope:

Rise =0 ; Run =Positive

Undefined Slope:

Rise=Positive or Negative; Run =0
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