line whose equation is to be
determine with the given line
y = mx + c are

yy m

m

− 11 = ± xx−

1

tan

tan

α()

∓ α^







   





A LINE EQUALLY INCLINED WITH TWO LINES
Let the two lines with slopes m 1 and m 2 be equally
inclined to a line with slope m

then mm
mm

mm

mm

1

1

2

(^112)
−

• =− −


• 
  

  ⎛
  ⎝⎜
  ⎞
  ⎠⎟^
  BISECTORS OF THE ANGLES BETWEEN TWO
  STRAIGHT LINES
  z Bisectors of the angles between the lines
  a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0
  (i) Containing the origin
  ax by c
  ab
  ax by c
  ab
  111
  1
  2
  1
  2
  222
  2
  2
  2
  2
  ++

  
  


• 
  

  = ++

  
  


• 
  

  (ii) Not containing the origin
  ax by c
  ab
  ax by c
  ab
  111
  12 12
  222
  22 22
  ++

  
  


• 
  

  =− ++

  
  


• 
  

  ()
  z To find the acute and obtuse angle bisectors
  Let θ be the angle between one of the lines and
  one of the bisectors given by
  ax by c
  ab
  ax by c
  ab
  111
  1
  2
  1
  2
  222
  2
  2
  2
  2
  ++

  
  


• 
  

  =±
  ++

  
  


• 
  

  .
  Find tanθ. If |tanθ| < 1, then this bisector is the
  bisector of acute angle and the other one is the
  bisector of the obtuse angle.
  If |tanθ| > 1, then this bisector is the bisector of
  obtuse angle and other one is the bisector of the
  acute angle.
  z Method to find acute angle bisector and obtuse
  angle bisector
  (i) I f a 1 a 2 + b 1 b 2 > 0, then the bisector
  corresponding to “+” sign gives the obtuse
  angle bisector and the bisector corresponding
  to “–” sign is the bisector of acute angle
  between the lines.
  
  
   
  
  
  
  
  
  
  
  
  
  
  
  (ii) I f a 1 a 2 + b 1 b 2 < 0, then the bisector
  corresponding to “+” and “–” sign given the
  acute and obtuse angle bisectors respectively.
  Remarks
  t
  #JTFDUPST
  BSF
  QFSQFOEJDVMBS
  UP
  FBDI
  PUIFS
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  *G
  a 1 a 2 + b 2 b 2 > 0, then the origin lies in obtuse
  angle and if a 1 a 2 + b 1 b 2 < 0, then the origin
  lies in acute angle.
  LENGTH OF PERPENDICULAR
  z Distance of a point from a line : The length p of
  the perpendicular from the point (x 1 , y 1 ) to the line
  ax + by + c = 0 is given by p
  ax by c
  ab
  = ++

  
  


• 
  

  || 11

  
  

22. 
    

    z Distance between two parallel lines : Let
    the two parallel lines be ax + by + c 1 = 0 and
    ax + by + c 2 = 0 then the distance between the lines
    is d
    ab

    
    

• λ
  ()^22
  , where
  (i) λ = |c 1 – c 2 |, if they are on the same side of
  origin.
  (ii) λ = |c 1 | + |c 2 |, if the origin O lies between
  them.
  POSITION OF A POINT WITH RESPECT TO A
  LINE
  Let the given line be ax + by + c = 0 and observing
  point is (x 1 , y 1 ), then
  z If the same sign is found by putting x = x 1 , y = y 1
  and x = 0, y = 0 in equation of line, then the point
  (x 1 , y 1 ) is situated on the side of the origin.
  z If the opposite sign is found by putting x = x 1 ,
  y = y 1 and x = 0, y = 0 in equation of line then the
  point (x 1 , y 1 ) is situated on the opposite side of the
  origin.
  CONCURRENT LINES
  Three or more lines are said to be concurrent lines if
  they meet at a point.
  z Three lines a 1 x + b 1 y + c 1 = 0, a 2 x + b 2 y + c 2 = 0 and
  a 3 x + b 3 y + c 3 = 0 are concurrent if,
  abc
  abc
  abc
  111
  222
  333
  = 0
  z The condition for the lines P = 0, Q = 0 and R = 0
  to be concurrent is that three constants a, b, c (not
  all zero at the same time) can be obtained such that
  aP + bQ + cR = 0