Remarks

t
 &RVBUJPO
PG
x-axis is

 









y = 0.

t
 &RVBUJPO
PG
B
MJOF


parallel to x-axis (or

perpendicular to

y-axis) at a distance of

‘b’ from it is y = b.

t
 &RVBUJPO
PG y-axis is

 









x = 0.
t
&RVBUJPO
PG
B
MJOF

parallel to y-axis (or
perpendicular to
x-axis) at a distance
of ‘a’ from it is x = a
EQUATION OF PARALLEL AND PERPENDICULAR
LINES TO A GIVEN LINE
z Equation of a line which is parallel to ax + by + c = 0
is ax + by + λ = 0
z Equation of a line which is perpendicular to
ax + by + c = 0 is bx – ay + λ = 0
where λ is an arbitrary constant.

GENERAL EQUATION OF A STRAIGHT LINE AND
ITS TRANSFORMATION IN STANDARD FORMS
General equation of a line is ax + by + c = 0. It can be
reduced in various standard forms given below.

z Slope intercept form: y

a

b

x c

b

=−−, slope m a

b

=−

and intercept on y-axis is, C c

b

=−

z Intercept form : x
ca

y

− cb

+

−

=

//

1 , where

x- intercept is ⎛⎝⎜−c⎞⎠⎟

a

and y-intercept is ⎜⎛⎝−c⎞⎠⎟

b

z Normal form : −

• 
  

−

+

=

+

ax

ab

by

ab

c

(^2222) ab 22
where cosα=−

• a
  ab^22
  , sinα=−


• 
  

  b
  ab^22
  and
  p c
  ab

  
  

  (^22) +
  POINT OF INTERSECTION OF TWO LINES
  Let a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0 be two
  non-parallel lines. If (x′, y′) be the co-ordinates of their
  point of intersection, then a 1 x′ + b 1 y′ + c 1 = 0 and
  a 2 x′ + b 2 y′ + c 2 = 0
  Solving these equations, we get
  (, )′′= − ,
  −
  −
  −
  ⎛
  ⎝⎜
  ⎞
  ⎠⎟
  xy bc b c
  ab a b
  ca c a
  ab a b
  12 21
  12 21
  12 21
  12 21

  
  

  ⎛
  ⎝
  ⎜
  ⎜
  ⎜
  ⎜
  ⎞
  ⎠
  ⎟
  ⎟
  ⎟
  ⎟
  bb
  cc
  aa
  bb
  cc
  aa
  aa
  bb
  12
  12
  12
  12
  12
  12
  12
  12
  ,
  GENERAL EQUATION OF LINES THROUGH THE
  INTERSECTION OF TWO GIVEN LINES
  If equation of two lines P : a 1 x + b 1 y + c 1 = 0 and
  Q : a 2 x + b 2 y + c 2 = 0, then the equation of the line
  passing through the intersection of these lines is
  P + λQ = 0 or a 1 x + b 1 y + c 1 + λ(a 2 x + b 2 y + c 2 ) = 0.
  ANGLE BETWEEN TWO NON-PARALLEL LINES
  z When equations are in slope intercept form
  Let θ be the angle between
  the lines y = m 1 x + c 1 and
  y = m 2 x + c 2
  ∴
  θ= −

  
  


• 
  

  tan−^112
  (^112)
  mm
  mm
  
  
     
   
   (^)  
   (^)  
  z When equations are in general form
  The angle θ between the lines a 1 x + b 1 y + c 1 = 0
  and a 2 x + b 2 y + c 2 = 0 is given by
  tanθ= −

  
  


• 
  

  ab ab
  aa bb
  21 12
  12 12
  .
  CONDITIONS FOR TWO LINES TO BE
  COINCIDENT, PARALLEL, PERPENDICULAR
  AND INTERSECTING
  z Two lines a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0
  are,
  (a) Coincident, if
  a
  a
  b
  b
  c
  c
  1
  2
  1
  2
  1
  2

  
  

  (b) Parallel, if a
  a
  b
  b
  c
  c
  1
  2
  1
  2
  1
  2
  = ≠
  (c) Intersecting, if a
  a
  b
  b
  1
  2
  1
  2
  ≠
  (d) Perpendicular, if a 1 a 2 + b 1 b 2 = 0
  EQUATION OF STRAIGHT LINES THROUGH A
  GIVEN POINT MAKING A GIVEN ANGLE WITH
  A GIVEN LINE
  Let P(x 1 , y 1 ) be a given point and y = mx + c be
  the given line. Let α be the angle made by that