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STRAIGHT LINES

DEFINITION: A straight line is defined as the curve which is such that the line segment joining any two points lies on it.
Every first degree equation in x, y represents a straight line equation.
i.e. ax + by + c = 0 (a2 + b2 ≠ 0) represents a straight line.
 If a line makes an angle 'θ' with the positive direction of the X − axis in anti clock wise sense then 'θ' is called inclination of a line. Where θ  [0 Π) and 'Tanθ' is defined as slope of the line. The slope of the line is denoted by 'm' (m = Tanθ)
 Since a line parallel to X − axis makes an angle of '0º' with the X − axis then its slope m = Tan0º = 0.
 Since a line parallel to Y − axis makes an angle of 90º with the X − axis then its slope m = Tan 90º = undefined (∞).
 Since if two lines parallel they have same inclination then their slopes are equal.
 If a line passing through two points A (x1, y1) and B(x2, y2) then its slope is 
If the 'θ' is acute angle between two lines having slopes 'm1' and 'm2' then


     
 If two lines parallel then θ = 0   m− m2 = 0  m1 = m2
 If two lines perpendicular then θ =    1 + m1 m2 = 0 
     m1m2 = −1


DIFFERENT FORMS OF THE EQUATION OF A LINE 

1) Line parallel to X − axis
 If a straight line parallel to X − axis and is at a distance of 'a' units from it an above the X − axis then its equation is y = a       

 If a straight line parallel to Y - axis and at a distance of 'a' units from it below the X - axis then its equation is y = −a. 

 
(2) Line parallel to Y - axis
 If a straight line parallel to Y − axis and at a distance of 'b' units from it on right side of Y − axis then its equation is x = b      

 If a straight line parallel to Y − axis and at a distance of 'b' units from it on left side of Y − axis then its equation is x = −b 

 

(3) Slope intercept form

 The equation of a line with slope 'm' and making an intercept 'c' on Y − axis is y = mx + c      

(4) point slope form
 The equation of a line passing through (x1, y1) and having slope 'm' is y − y1 = m(x − x1)

 

(5) Two point form of a line
 The equation of a line passing through (x1, y1) and (x2, y2) is

 

      
(6) The intercept form of a line
 The equation of the straight line which cuts off intercepts of lengths of 'a' and 'b' on X − axis and Y − axis respectively is  
Where 'a' is called x − intercept and 'b' is called y − intercept.
Where the line cuts X − axis at (a, 0) and Y − axis at (0, b)

(7) Normal form of a line
The equation of straight line which is at a distance of 'p' units from the origin   and it's normal (OP) makes an angle 'α' (0 α 2Π)  with positive direction of X − axis is x cosα + y sin α = p where p > 0 and 0 α

 

REDUCTION OF GENERAL EQUATION TO STANDARD FORM 

General equation of straight line Ax + By + C = 0 (A2 + B2 ≠ 0)
 

(1) Reduction to 'Slope-Intercept' form
       If B≠0 then Ax + By + C = 0 can be written as 
         
       Compare (i) with y = mx+c then slope m = 
        Y intercept c = 

 

(2) Reduction to 'Intercept' form
       If C ≠ 0 then Ax + By + C = 0 can be written as


        
       Compare (ii) with    then
       x − intercept a =  ; y − intercept b = 

 

(3) Reduction to 'Normal' form
Let Ax + By + C = 0 can be written as Ax + By = −C ................ (iii)
compare (iii ) with xcosα + ysinα = P then we get


    
Here we have to choose the signs of cosα, sinα so that 'P' should be + ve

 

PARAMETRIC FORM OF A LINE 

The equation of a straight line passing through (x1, y1) and having inclination θ is

    = r.
Where 'r' is distance of the point (x, y) on the line from the point (x1, y1).
 The co-ordinates of the points on the line  
which are at a distance of 'r' units from the point (x1, y1) is (x1 ± r cosθ, y1 ± rsinθ)


 

 Equation of a line which is parallel to the given line ax + by + c = 0 is ax + by + k = 0 (k is some constant)
 Equation of a line which is perpendicular to the given line ax + by + c = 0 is bx − ay + λ = 0 (λ is some constant).

 

POSITION OF TWO POINTS RELATIVE TO A LINE 

 The points (x1, y1) and (x2, y2) lie on the same side or opposite sides of the given line ax + by + c = 0 according as (ax1 + by1 + c) (ax2 + by2 + c) > 0 or < 0 respectively.
 The point (x1, y1) will lies on origin side of the line ax + by + c = 0 if (ax+ by+ c) c > 0
 The point (x1, y1) will lies on non-origin side of the line ax + by + c = 0 if (ax+ by+ c) c < 0

 

POINT OF INTERSECTION OF TWO LINES 

Point of intersection of the lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 is


      
 Any line through the point of intersection of the lines a1x + b1y + c= 0 and a2x + b2y + c2 = 0 is of the form a1x + b1y + c1 + λ (a2x + b2y + c2) = 0 where λ  R

 

 DISTANCE FROM A POINT TO A LINE

 The distance (perpendicular distance) from a point (x1, y1) to a line ax + by + c = 0 is


         
 The perpendicular distance from origin to a line ax + by + c = 0 is    

 The distance between two parallel lines ax + by + c= 0 and ax + by + c2 = 0 is        
                     
 EQUATIONS OF STRAIGHT LINES PASSING THROUGH A GIVEN POINT AND MAKING A GIVEN ANGLE WITH A GIVEN LINE

 Equation of straight lines passing through the given point (x1, y1) and making given angle 'α' with the given line y = mx + c is y − y1 = Tan (θ ± α) (x − x1) where Tan θ = m

 

EQUATIONS OF THE BISECTORS 

The equations of the bisectors of the angles between the lines
a1x + b1y + c1 = 0 ........... (1) and
a2x + b2y + c2 = 0 ........... (2) is

 
      
EQUATION OF THE BISECTORS OF THE ANGLE CONTAINING THE ORIGIN AND NOT CONTAINING THE ORIGIN 

 Equation of the bisector of the angle between the lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 (where c1 > 0 and c2 > 0) containing the origin is
   

 Equation of the bisector of the angle between the lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 (where c> 0 and c> 0) not containing the origin is


    .
 The equation of the bisector of the angle between the two lines a1x + b1y + c1 = 0 and a1x + b2y + c2 = 0 containing the point (h, k) will be


    .
    
   a1h + b1k + c1 and a2h + b2k + c2 are of the same sign or opposite sign
 Let the equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 where c1 > 0; c2 > 0 and a1b2 ≠ a2b1
the equations bisectors are   

(i) If (a1a2 + b1b2) > 0 then '+' sign gives obtuse angular bisector and '−' sign gives acute angular bisector.
(ii) (a1a2 + b1b2) < 0 then '+' sign gives acute angular bisector and '−' sign gives obtuse angular bisector.
 Angular bisectors are perpendicular to each other.
 If (a1a2 + b1b2) > 0 then origin lies in obtuse angle.
If (a1a2 + b1b2) < 0 then origin lies in acute angle.
The foot of the perpendicular drawn from the point (x1, y1) to the line ax + by + c = 0 

If (h, k) is foot of the perpendicular drawn from the point (x1, y1) to the line  ax + by + c = 0 then   
Image of a point (x1, y1) about a line ax + by + c = 0 

If (h, k) is the image of a point (x1, y1) about the line ax + by + c = 0
 then  

 The image of a point (α, β) with respect to X − axis is (α, − β)

 The image of a point (α, β) with respect to Y − axis is (−α, β)

 The image of a point (α, β) with respect to origin is (−α, −β)

 The image of a point (α, β) with respect to the line x = a is (2a − α, β)

 The image of a point (α, β) with respect to the line y = b is (α, 2b − β)

 The image of a point (α, β) with respect to the line y = x is (β, α)

Posted Date : 19-02-2021

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