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Slope of a Line

Slope (or gradient) of a line is defined as the tangent of the angle which a line makes with +ve X-axis. It is denoted by m.    m = tan

 

 
Note:
1.  m can be defined as tan  for    0     &       ≠ /2.
2.  the slope of a line parallel to X-axis = 0 perpondicular to X-axis is undefined.

 

Slope of a line passing through two given points:

Let the two points be A () and B ().

 

Slope of line L = m = tan

ð        () 
Acute angle between two lines  of given slope

Slope of

Slope of

 

If  is the acute angle between  then:

tan  =   

 

If is parallel to        

 

 

If is perpendicular to        

 

Intercepts of a Line

The line L cuts X and Y axes in A and respectively.

X-intercept of L = OA = a

Y-intercept of L = OB= b


X-intercept will be negative if L intersects –ve X-axis and Y-intercept will be negative if L intersects –ve Y-axis

 

If L is parallel to X-axis, X-intercept is undefined and

If L is parallel to Y-axis, Y-intercept is undefined.

 

Equation of a line in various forms

1.  Equation of line L of slope m and cutting off an intercept b on Y-axis:

Any point P (x, y) on line L satisfies the following:                 

tan  = m =  

       

ð   is the equation line L.      

This is also known as slope – intercept form. 
2.  Equation of line L of slope m and passing through a given point:

 

Any point P(x, y) on line L satisfies the following:

Slope = m =  

 

       is the equation of line L.

 

This also known as point – slope form. 
3.  Equation of line L passing through two given point:

Any point P(x, y) on line L satisfies the following:

Slope of PA =     & slope of AB =

 

As PAB is straight line:

      slope of (PA) = slope of (PAB)

 

           

 

        is the equation of line .

 

In the determinant form it is given as:

     = 0    

 


This is also known as two – point form.





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