Straight Lines Class 11 Notes Maths Chapter 9

 Page 1


STRAIGHT LINE
1. INTRODUCTION
Co-ordinate geometry is the branch of mathematics which includes the study of different curves and figures by 
ordered pairs of real numbers called Cartesian co-ordinates, representing lines & curves by algebraic equation. This 
mathematical model is used in solving real world problems.
2. CO-ORDINATE SYSTEM
Co-ordinate system is nothing but a reference system designed to locate position of any point or geometric 
element in a plane of space.
2.1 Cartesian Co-ordinates
Let us consider two perpendicular straight lines XOX’ and YOY’ passing through the origin  
y
x’ O
y’
x
  =90
Figure 8.1
O in the plane. Then, 
Axis of x: The horizontal line xox’ is called axis of x.
Axis of y: The vertical line yoy’ is called axis of y.
Co-ordinate axis: x-axis and y-axis together are called axis of co-ordinates or axis of 
reference.
Origin: The point ‘O’ is called the origin of co-ordinates or just the origin. 
Oblique axis: When xox’ and yoy’ are not at right angle, i.e. if the both axes are not perpendicular, to each other, 
then axis of co-ordinates are called oblique axis. 
2.2 Co-ordinate of a Point
The ordered pair of perpendicular distances of a point from X- and Y-axes are called co-ordinates of that point.
If the perpendicular algebraic distance of a point p from y-axis is x and from x-axis is y, then co-ordinates of the 
point P is (x, y). Here,
(a) x is called x-co-ordinate or abscissa.
(b) y is called y-co-ordinate or ordinate.
(c) x-co-ordinate of every point lying upon y-axis is zero.
(d) y-co-ordinate of every point lying upon x-axis is zero.
(e) Co-ordinates of origin are (0, 0).
Note: A point whose abscissa and ordinate are both integers is known as lattice point. 
Page 2


STRAIGHT LINE
1. INTRODUCTION
Co-ordinate geometry is the branch of mathematics which includes the study of different curves and figures by 
ordered pairs of real numbers called Cartesian co-ordinates, representing lines & curves by algebraic equation. This 
mathematical model is used in solving real world problems.
2. CO-ORDINATE SYSTEM
Co-ordinate system is nothing but a reference system designed to locate position of any point or geometric 
element in a plane of space.
2.1 Cartesian Co-ordinates
Let us consider two perpendicular straight lines XOX’ and YOY’ passing through the origin  
y
x’ O
y’
x
  =90
Figure 8.1
O in the plane. Then, 
Axis of x: The horizontal line xox’ is called axis of x.
Axis of y: The vertical line yoy’ is called axis of y.
Co-ordinate axis: x-axis and y-axis together are called axis of co-ordinates or axis of 
reference.
Origin: The point ‘O’ is called the origin of co-ordinates or just the origin. 
Oblique axis: When xox’ and yoy’ are not at right angle, i.e. if the both axes are not perpendicular, to each other, 
then axis of co-ordinates are called oblique axis. 
2.2 Co-ordinate of a Point
The ordered pair of perpendicular distances of a point from X- and Y-axes are called co-ordinates of that point.
If the perpendicular algebraic distance of a point p from y-axis is x and from x-axis is y, then co-ordinates of the 
point P is (x, y). Here,
(a) x is called x-co-ordinate or abscissa.
(b) y is called y-co-ordinate or ordinate.
(c) x-co-ordinate of every point lying upon y-axis is zero.
(d) y-co-ordinate of every point lying upon x-axis is zero.
(e) Co-ordinates of origin are (0, 0).
Note: A point whose abscissa and ordinate are both integers is known as lattice point. 
MASTERJEE CONCEPTS
2.3 Polar Co-ordinates
Let OX be any fixed line, known as initial line, and O be the origin. If the distance of any      
y
x
r
O
P (r , ) 

Figure 8.2
point P from the origin O is ‘r’ and ?XOP = ?, then (r, ?) are known as polar co-ordinates  
of point P . If (x, y) are the Cartesian co-ordinates of a point P , then x = rcos ?; y = rsin ? and 
| r | =
2 2
x y + , ? = tan
–1
 
y
x
? ?
? ?
? ?
 
( )
, ? ? -p p
Illustration 1: If the Cartesian co-ordinates of any point are ( 3,1) , find the polar co-ordinates.  (JEE MAIN)
Sol: Polar co-ordinates of any point are (r, ?), where r =
2 2
x y + and ? = tan
–1
 
y
x
? ?
? ?
? ?
.
 x 3 = ; y = 1
Let their polar co-ordinates be (r, ?) ? x = r cos ?; y = r sin? 
So r ? 
2 2
x y +   r = 3 1 +
  
? ? 
1
y
tan
x
-
? ?
? ?
? ?
 = 2   ? ? 
1
1
tan
6
3
-
? ?
p
=
? ?
? ?
 
?(r, ?) = 2,
6
? ? p
? ?
? ?
.
3. DISTANCE FORMULA
The distance between two points P(x
1
, y
1
) and Q(x
2
, y
2
) is          
y
Q(x ,y )
2 2
d
P(x ,y )
1 1
O
x - x
2 1
x
2
A
B(r , )
2 2

(r , )
1 1

Figure 8.3
PQ = 
2 2
1 2 1 2
(x –x ) (y – y ) + = 
- + -
2 2
2 1 2 1
(x x ) (y y )
Distance of a point P(x
1
, y
1
) from the origin O(0, 0) is
OP = 
2 2
1 1
X y +
 
Distance between two polar co-ordinates A(r
1
, ?
1
) and B(r
2
, ?
2
) is 
given by 
AB = 
2 2
1 2 1 2 1 2
r r 2r r cos( ) + - ? - ?
Proof: AB = 
2 2
2 1 2 1
(x x ) (y y ) - + - x
1
 = r
1
 cos ?
1
, x
2
 = r
2
 cos ?
2
, y
1
 = r
1
sin ?
1
, y
2
 = r
2
sin?
2
  
AB = 
2 2
2 2 1 1 2 2 1 1
(r cos r cos ) (r sin r sin ) ? - ? + ? - ?
AB = 
2 2 2 2
2 2 1 2 1 2 1 1 2 2 12 1 2 1 1
(r cos ) 2rr cos cos (r cos ) (r sin ) 2rr sin sin (r sin ) ? - ? ? + ? + ? - ? ? + ?
AB = 
2 2
1 2 1 2 1 2
r r 2r r cos( ) + - ? - ? 
Distance between two polar co-ordinates A(r
1
, q
1
) and B(r
2
, q
2
) is given by
AB = 
2 2
1 2 1 2 1 2
r r 2r r cos( ) + - ? - ? 
Vaibhav Krishnan (JEE 2009, AIR 22)
Page 3


STRAIGHT LINE
1. INTRODUCTION
Co-ordinate geometry is the branch of mathematics which includes the study of different curves and figures by 
ordered pairs of real numbers called Cartesian co-ordinates, representing lines & curves by algebraic equation. This 
mathematical model is used in solving real world problems.
2. CO-ORDINATE SYSTEM
Co-ordinate system is nothing but a reference system designed to locate position of any point or geometric 
element in a plane of space.
2.1 Cartesian Co-ordinates
Let us consider two perpendicular straight lines XOX’ and YOY’ passing through the origin  
y
x’ O
y’
x
  =90
Figure 8.1
O in the plane. Then, 
Axis of x: The horizontal line xox’ is called axis of x.
Axis of y: The vertical line yoy’ is called axis of y.
Co-ordinate axis: x-axis and y-axis together are called axis of co-ordinates or axis of 
reference.
Origin: The point ‘O’ is called the origin of co-ordinates or just the origin. 
Oblique axis: When xox’ and yoy’ are not at right angle, i.e. if the both axes are not perpendicular, to each other, 
then axis of co-ordinates are called oblique axis. 
2.2 Co-ordinate of a Point
The ordered pair of perpendicular distances of a point from X- and Y-axes are called co-ordinates of that point.
If the perpendicular algebraic distance of a point p from y-axis is x and from x-axis is y, then co-ordinates of the 
point P is (x, y). Here,
(a) x is called x-co-ordinate or abscissa.
(b) y is called y-co-ordinate or ordinate.
(c) x-co-ordinate of every point lying upon y-axis is zero.
(d) y-co-ordinate of every point lying upon x-axis is zero.
(e) Co-ordinates of origin are (0, 0).
Note: A point whose abscissa and ordinate are both integers is known as lattice point. 
MASTERJEE CONCEPTS
2.3 Polar Co-ordinates
Let OX be any fixed line, known as initial line, and O be the origin. If the distance of any      
y
x
r
O
P (r , ) 

Figure 8.2
point P from the origin O is ‘r’ and ?XOP = ?, then (r, ?) are known as polar co-ordinates  
of point P . If (x, y) are the Cartesian co-ordinates of a point P , then x = rcos ?; y = rsin ? and 
| r | =
2 2
x y + , ? = tan
–1
 
y
x
? ?
? ?
? ?
 
( )
, ? ? -p p
Illustration 1: If the Cartesian co-ordinates of any point are ( 3,1) , find the polar co-ordinates.  (JEE MAIN)
Sol: Polar co-ordinates of any point are (r, ?), where r =
2 2
x y + and ? = tan
–1
 
y
x
? ?
? ?
? ?
.
 x 3 = ; y = 1
Let their polar co-ordinates be (r, ?) ? x = r cos ?; y = r sin? 
So r ? 
2 2
x y +   r = 3 1 +
  
? ? 
1
y
tan
x
-
? ?
? ?
? ?
 = 2   ? ? 
1
1
tan
6
3
-
? ?
p
=
? ?
? ?
 
?(r, ?) = 2,
6
? ? p
? ?
? ?
.
3. DISTANCE FORMULA
The distance between two points P(x
1
, y
1
) and Q(x
2
, y
2
) is          
y
Q(x ,y )
2 2
d
P(x ,y )
1 1
O
x - x
2 1
x
2
A
B(r , )
2 2

(r , )
1 1

Figure 8.3
PQ = 
2 2
1 2 1 2
(x –x ) (y – y ) + = 
- + -
2 2
2 1 2 1
(x x ) (y y )
Distance of a point P(x
1
, y
1
) from the origin O(0, 0) is
OP = 
2 2
1 1
X y +
 
Distance between two polar co-ordinates A(r
1
, ?
1
) and B(r
2
, ?
2
) is 
given by 
AB = 
2 2
1 2 1 2 1 2
r r 2r r cos( ) + - ? - ?
Proof: AB = 
2 2
2 1 2 1
(x x ) (y y ) - + - x
1
 = r
1
 cos ?
1
, x
2
 = r
2
 cos ?
2
, y
1
 = r
1
sin ?
1
, y
2
 = r
2
sin?
2
  
AB = 
2 2
2 2 1 1 2 2 1 1
(r cos r cos ) (r sin r sin ) ? - ? + ? - ?
AB = 
2 2 2 2
2 2 1 2 1 2 1 1 2 2 12 1 2 1 1
(r cos ) 2rr cos cos (r cos ) (r sin ) 2rr sin sin (r sin ) ? - ? ? + ? + ? - ? ? + ?
AB = 
2 2
1 2 1 2 1 2
r r 2r r cos( ) + - ? - ? 
Distance between two polar co-ordinates A(r
1
, q
1
) and B(r
2
, q
2
) is given by
AB = 
2 2
1 2 1 2 1 2
r r 2r r cos( ) + - ? - ? 
Vaibhav Krishnan (JEE 2009, AIR 22)
Illustration 2: Find the distance between P 2,
6
? ? p
-
? ?
? ?
 and Q 3,
6
? ? p
? ?
? ?
.    (JEE MAIN) 
Sol: The distance between two points = 
2 2
1 2 12 1 2
r r 2r r cos( ) + - ? -?
.
 Therefore,
 PQ =
2 2
1 2 12 1 2
1
r r 2r r cos( ) 4 9 2.2.3cos 4 9 12cos 13 12. 7
66 3 2
? ? ?? pp p
+ - ?-? = +- - - == +- = - =
? ? ??
? ? ??
Illustration 3: The point whose abscissa is equal to its ordinate and which is equidistant from the points A(1, 0), 
B(0, 3) is   (JEE MAIN) 
Sol: Given, abscissa = ordinate. Therefore distance can be found by considering the co-ordinates of required point 
be P(k, k).
Now given PA = PB ? 
2 2
(k 1) k - + = 
2 2
k (k 3) + -
2k
2
 – 2k + 1 = 2k
2
 – 6k + 9 ? 4k = 8 ? k = 2   
4. SECTION FORMULA
Let R divide the two points P(x
1
, y
1
) and Q(x
2
, y
2
) internally in the ratio m:n. 
Let (x, y) be the co-ordinates of R.
Draw PM, QN, RK perpendicular to the x-axis. 
Also, draw PE and RF perpendicular to RK and QN.
Here, 
PR m
RQ n
= .
Triangles PRE and RFQ are similar. 
? 
PR PE
RQ RF
= ? 
PE m
RF n
=
But PE = x – x
1 
and RF = x
2
 – x.  
? 
1
2
xx m
x x n
-
=
-
 ? 
2 1
mx nx
x
m n
+
=
+
In the same way, 
ER m
FQ n
= 
i.e., 
1
2
y y m
y y n
-
=
-
? y = 
2 1
my ny
m n
+
+
The co-ordinates of R are 
2 1 2 1
mx nx my ny
,
m n m n
? ? + +
? ?
? ?
+ +
? ?
If R’ divides PQ externally, so that 
PR' m
QR' n
= , triangles PER’ and QR’F are similar.  
Figure 8.5
y
x’ O
y’
x
M N
K
P
Q
R’ F
E
? 
PR' PE
R'Q R'F
=
            
But PE = x – x
1 
and R’F = x – x
2
. 
? 
1
2
xx m
xx n
-
=
-
   i.e., x = 
2 1
mx nx
mn
-
-
Similarly, 
2 1
my ny
y
mn
-
=
-
.
The co-ordinates of R’ are 
2 1 2 1
mx nx my ny
,
mn mn
? ? - -
? ?
? ?
- -
? ?
 
y
x’
y’
x
P
R
Q
E
F
O
M N K
Figure 8.4
Page 4


STRAIGHT LINE
1. INTRODUCTION
Co-ordinate geometry is the branch of mathematics which includes the study of different curves and figures by 
ordered pairs of real numbers called Cartesian co-ordinates, representing lines & curves by algebraic equation. This 
mathematical model is used in solving real world problems.
2. CO-ORDINATE SYSTEM
Co-ordinate system is nothing but a reference system designed to locate position of any point or geometric 
element in a plane of space.
2.1 Cartesian Co-ordinates
Let us consider two perpendicular straight lines XOX’ and YOY’ passing through the origin  
y
x’ O
y’
x
  =90
Figure 8.1
O in the plane. Then, 
Axis of x: The horizontal line xox’ is called axis of x.
Axis of y: The vertical line yoy’ is called axis of y.
Co-ordinate axis: x-axis and y-axis together are called axis of co-ordinates or axis of 
reference.
Origin: The point ‘O’ is called the origin of co-ordinates or just the origin. 
Oblique axis: When xox’ and yoy’ are not at right angle, i.e. if the both axes are not perpendicular, to each other, 
then axis of co-ordinates are called oblique axis. 
2.2 Co-ordinate of a Point
The ordered pair of perpendicular distances of a point from X- and Y-axes are called co-ordinates of that point.
If the perpendicular algebraic distance of a point p from y-axis is x and from x-axis is y, then co-ordinates of the 
point P is (x, y). Here,
(a) x is called x-co-ordinate or abscissa.
(b) y is called y-co-ordinate or ordinate.
(c) x-co-ordinate of every point lying upon y-axis is zero.
(d) y-co-ordinate of every point lying upon x-axis is zero.
(e) Co-ordinates of origin are (0, 0).
Note: A point whose abscissa and ordinate are both integers is known as lattice point. 
MASTERJEE CONCEPTS
2.3 Polar Co-ordinates
Let OX be any fixed line, known as initial line, and O be the origin. If the distance of any      
y
x
r
O
P (r , ) 

Figure 8.2
point P from the origin O is ‘r’ and ?XOP = ?, then (r, ?) are known as polar co-ordinates  
of point P . If (x, y) are the Cartesian co-ordinates of a point P , then x = rcos ?; y = rsin ? and 
| r | =
2 2
x y + , ? = tan
–1
 
y
x
? ?
? ?
? ?
 
( )
, ? ? -p p
Illustration 1: If the Cartesian co-ordinates of any point are ( 3,1) , find the polar co-ordinates.  (JEE MAIN)
Sol: Polar co-ordinates of any point are (r, ?), where r =
2 2
x y + and ? = tan
–1
 
y
x
? ?
? ?
? ?
.
 x 3 = ; y = 1
Let their polar co-ordinates be (r, ?) ? x = r cos ?; y = r sin? 
So r ? 
2 2
x y +   r = 3 1 +
  
? ? 
1
y
tan
x
-
? ?
? ?
? ?
 = 2   ? ? 
1
1
tan
6
3
-
? ?
p
=
? ?
? ?
 
?(r, ?) = 2,
6
? ? p
? ?
? ?
.
3. DISTANCE FORMULA
The distance between two points P(x
1
, y
1
) and Q(x
2
, y
2
) is          
y
Q(x ,y )
2 2
d
P(x ,y )
1 1
O
x - x
2 1
x
2
A
B(r , )
2 2

(r , )
1 1

Figure 8.3
PQ = 
2 2
1 2 1 2
(x –x ) (y – y ) + = 
- + -
2 2
2 1 2 1
(x x ) (y y )
Distance of a point P(x
1
, y
1
) from the origin O(0, 0) is
OP = 
2 2
1 1
X y +
 
Distance between two polar co-ordinates A(r
1
, ?
1
) and B(r
2
, ?
2
) is 
given by 
AB = 
2 2
1 2 1 2 1 2
r r 2r r cos( ) + - ? - ?
Proof: AB = 
2 2
2 1 2 1
(x x ) (y y ) - + - x
1
 = r
1
 cos ?
1
, x
2
 = r
2
 cos ?
2
, y
1
 = r
1
sin ?
1
, y
2
 = r
2
sin?
2
  
AB = 
2 2
2 2 1 1 2 2 1 1
(r cos r cos ) (r sin r sin ) ? - ? + ? - ?
AB = 
2 2 2 2
2 2 1 2 1 2 1 1 2 2 12 1 2 1 1
(r cos ) 2rr cos cos (r cos ) (r sin ) 2rr sin sin (r sin ) ? - ? ? + ? + ? - ? ? + ?
AB = 
2 2
1 2 1 2 1 2
r r 2r r cos( ) + - ? - ? 
Distance between two polar co-ordinates A(r
1
, q
1
) and B(r
2
, q
2
) is given by
AB = 
2 2
1 2 1 2 1 2
r r 2r r cos( ) + - ? - ? 
Vaibhav Krishnan (JEE 2009, AIR 22)
Illustration 2: Find the distance between P 2,
6
? ? p
-
? ?
? ?
 and Q 3,
6
? ? p
? ?
? ?
.    (JEE MAIN) 
Sol: The distance between two points = 
2 2
1 2 12 1 2
r r 2r r cos( ) + - ? -?
.
 Therefore,
 PQ =
2 2
1 2 12 1 2
1
r r 2r r cos( ) 4 9 2.2.3cos 4 9 12cos 13 12. 7
66 3 2
? ? ?? pp p
+ - ?-? = +- - - == +- = - =
? ? ??
? ? ??
Illustration 3: The point whose abscissa is equal to its ordinate and which is equidistant from the points A(1, 0), 
B(0, 3) is   (JEE MAIN) 
Sol: Given, abscissa = ordinate. Therefore distance can be found by considering the co-ordinates of required point 
be P(k, k).
Now given PA = PB ? 
2 2
(k 1) k - + = 
2 2
k (k 3) + -
2k
2
 – 2k + 1 = 2k
2
 – 6k + 9 ? 4k = 8 ? k = 2   
4. SECTION FORMULA
Let R divide the two points P(x
1
, y
1
) and Q(x
2
, y
2
) internally in the ratio m:n. 
Let (x, y) be the co-ordinates of R.
Draw PM, QN, RK perpendicular to the x-axis. 
Also, draw PE and RF perpendicular to RK and QN.
Here, 
PR m
RQ n
= .
Triangles PRE and RFQ are similar. 
? 
PR PE
RQ RF
= ? 
PE m
RF n
=
But PE = x – x
1 
and RF = x
2
 – x.  
? 
1
2
xx m
x x n
-
=
-
 ? 
2 1
mx nx
x
m n
+
=
+
In the same way, 
ER m
FQ n
= 
i.e., 
1
2
y y m
y y n
-
=
-
? y = 
2 1
my ny
m n
+
+
The co-ordinates of R are 
2 1 2 1
mx nx my ny
,
m n m n
? ? + +
? ?
? ?
+ +
? ?
If R’ divides PQ externally, so that 
PR' m
QR' n
= , triangles PER’ and QR’F are similar.  
Figure 8.5
y
x’ O
y’
x
M N
K
P
Q
R’ F
E
? 
PR' PE
R'Q R'F
=
            
But PE = x – x
1 
and R’F = x – x
2
. 
? 
1
2
xx m
xx n
-
=
-
   i.e., x = 
2 1
mx nx
mn
-
-
Similarly, 
2 1
my ny
y
mn
-
=
-
.
The co-ordinates of R’ are 
2 1 2 1
mx nx my ny
,
mn mn
? ? - -
? ?
? ?
- -
? ?
 
y
x’
y’
x
P
R
Q
E
F
O
M N K
Figure 8.4
Alternate Method: 
PR' m m
R'Q n –n
= - =
By changing n into –n in the co-ordinates of R, we can obtain the co-ordinates 
of R’: 
2 1 2 1
mx nx my ny
,
mn mn
- -
- -
 
Cor. The mid-point joining the two points (x
1
, y
1
) and (x
2
, y
2
) is 
1 2 1 2
x xy y
,
2 2
? ? + +
? ?
? ?
? ? .
Cor. From the above cor., the co-ordinates of a point dividing PQ in the ratio ?:1 are 
1 2 1 2
x xy y
,
1 1
? ? + ? + ?
? ?
+ ? + ? ? ?
. Considering 
? as a variable parameter, i.e. of all values positive or negative, the co-ordinates of any point on the line joining the 
points (x
1
, y
1
) and (x
2
, y
2
) can be expressed in the above forms.
5. SPECIAL POINTS OF A TRIANGLE 
5.1 Centroid
Let the vertices of the triangle ABC be (x
1
, y
1
), (x
2
, y
2
) and (x
3
, y
3
), respectively.  
The mid-point D of BC is 
2 3 2 3
x xy y
,
2 2
? ? + +
? ?
? ?
? ?
G, the centroid, divides AD internally 
in the ratio 2:1. 
Let G be (x, y),
then x = 
( )
2 3 1
2. (x x )/2 1.x
21
+ +
+
 = 
1 2 3
x x x
3
+ +
 and
( )
2 3 1
2. (y y )/2 1.y
y
21
+ +
=
+
= 
1 2 3
y y y
3
++
? G is 
1 2 3 1 2 3
x x xy y y
,
3 3
? ? + + + +
? ?
? ?
? ?
.
5.2 Incentre
Let A (x
1
, y
1
), B (x
2
, y
2
), C (x
3
, y
3
) be the vertices of the triangle. 
Let AD bisect angle BAC and cut BC at D.
We know that 
BD AB c
DC AC b
= =
Hence the co-ordinates of D are 
3 2 3 2
cx bx cy by
,
c b c b
+ +
+ +
                      
A
B
C
D
I
Figure 8.7
Let (x, y) be the incentre of the triangle
CD b
BD c
= ? 
BC b c
DB c
+
= ? BD = 
ca
bc +
( )
AI AB c b c
ID BD a ca/(b c)
+
= = =
+
? 
( )
( )
3 2 1
1 2 3
3 2 1
1 2 3
(b c) (cx bx )/(c b) ax
ax bx cx
x ,
bc a a bc
(b c) (cy by )/(c b) ay
ay by cy
y
bc a a bc
+ + + +
+ +
= =
++ + +
+ + ++
+ +
= =
++ + +
( )
( )
3 2 1
1 2 3
3 2 1
1 2 3
(b c) (cx bx )/(c b) ax
ax bx cx
x ,
bc a a bc
(b c) (cy by )/(c b) ay
ay by cy
y
bc a a bc
+ + + +
+ +
= =
++ + +
+ + ++
+ +
= =
++ + +
Figure 8.6
C(x , y)
3 3
F
D
G
2
1
E (x , y)
1 1
A B(x , y)
2 2
Page 5


STRAIGHT LINE
1. INTRODUCTION
Co-ordinate geometry is the branch of mathematics which includes the study of different curves and figures by 
ordered pairs of real numbers called Cartesian co-ordinates, representing lines & curves by algebraic equation. This 
mathematical model is used in solving real world problems.
2. CO-ORDINATE SYSTEM
Co-ordinate system is nothing but a reference system designed to locate position of any point or geometric 
element in a plane of space.
2.1 Cartesian Co-ordinates
Let us consider two perpendicular straight lines XOX’ and YOY’ passing through the origin  
y
x’ O
y’
x
  =90
Figure 8.1
O in the plane. Then, 
Axis of x: The horizontal line xox’ is called axis of x.
Axis of y: The vertical line yoy’ is called axis of y.
Co-ordinate axis: x-axis and y-axis together are called axis of co-ordinates or axis of 
reference.
Origin: The point ‘O’ is called the origin of co-ordinates or just the origin. 
Oblique axis: When xox’ and yoy’ are not at right angle, i.e. if the both axes are not perpendicular, to each other, 
then axis of co-ordinates are called oblique axis. 
2.2 Co-ordinate of a Point
The ordered pair of perpendicular distances of a point from X- and Y-axes are called co-ordinates of that point.
If the perpendicular algebraic distance of a point p from y-axis is x and from x-axis is y, then co-ordinates of the 
point P is (x, y). Here,
(a) x is called x-co-ordinate or abscissa.
(b) y is called y-co-ordinate or ordinate.
(c) x-co-ordinate of every point lying upon y-axis is zero.
(d) y-co-ordinate of every point lying upon x-axis is zero.
(e) Co-ordinates of origin are (0, 0).
Note: A point whose abscissa and ordinate are both integers is known as lattice point. 
MASTERJEE CONCEPTS
2.3 Polar Co-ordinates
Let OX be any fixed line, known as initial line, and O be the origin. If the distance of any      
y
x
r
O
P (r , ) 

Figure 8.2
point P from the origin O is ‘r’ and ?XOP = ?, then (r, ?) are known as polar co-ordinates  
of point P . If (x, y) are the Cartesian co-ordinates of a point P , then x = rcos ?; y = rsin ? and 
| r | =
2 2
x y + , ? = tan
–1
 
y
x
? ?
? ?
? ?
 
( )
, ? ? -p p
Illustration 1: If the Cartesian co-ordinates of any point are ( 3,1) , find the polar co-ordinates.  (JEE MAIN)
Sol: Polar co-ordinates of any point are (r, ?), where r =
2 2
x y + and ? = tan
–1
 
y
x
? ?
? ?
? ?
.
 x 3 = ; y = 1
Let their polar co-ordinates be (r, ?) ? x = r cos ?; y = r sin? 
So r ? 
2 2
x y +   r = 3 1 +
  
? ? 
1
y
tan
x
-
? ?
? ?
? ?
 = 2   ? ? 
1
1
tan
6
3
-
? ?
p
=
? ?
? ?
 
?(r, ?) = 2,
6
? ? p
? ?
? ?
.
3. DISTANCE FORMULA
The distance between two points P(x
1
, y
1
) and Q(x
2
, y
2
) is          
y
Q(x ,y )
2 2
d
P(x ,y )
1 1
O
x - x
2 1
x
2
A
B(r , )
2 2

(r , )
1 1

Figure 8.3
PQ = 
2 2
1 2 1 2
(x –x ) (y – y ) + = 
- + -
2 2
2 1 2 1
(x x ) (y y )
Distance of a point P(x
1
, y
1
) from the origin O(0, 0) is
OP = 
2 2
1 1
X y +
 
Distance between two polar co-ordinates A(r
1
, ?
1
) and B(r
2
, ?
2
) is 
given by 
AB = 
2 2
1 2 1 2 1 2
r r 2r r cos( ) + - ? - ?
Proof: AB = 
2 2
2 1 2 1
(x x ) (y y ) - + - x
1
 = r
1
 cos ?
1
, x
2
 = r
2
 cos ?
2
, y
1
 = r
1
sin ?
1
, y
2
 = r
2
sin?
2
  
AB = 
2 2
2 2 1 1 2 2 1 1
(r cos r cos ) (r sin r sin ) ? - ? + ? - ?
AB = 
2 2 2 2
2 2 1 2 1 2 1 1 2 2 12 1 2 1 1
(r cos ) 2rr cos cos (r cos ) (r sin ) 2rr sin sin (r sin ) ? - ? ? + ? + ? - ? ? + ?
AB = 
2 2
1 2 1 2 1 2
r r 2r r cos( ) + - ? - ? 
Distance between two polar co-ordinates A(r
1
, q
1
) and B(r
2
, q
2
) is given by
AB = 
2 2
1 2 1 2 1 2
r r 2r r cos( ) + - ? - ? 
Vaibhav Krishnan (JEE 2009, AIR 22)
Illustration 2: Find the distance between P 2,
6
? ? p
-
? ?
? ?
 and Q 3,
6
? ? p
? ?
? ?
.    (JEE MAIN) 
Sol: The distance between two points = 
2 2
1 2 12 1 2
r r 2r r cos( ) + - ? -?
.
 Therefore,
 PQ =
2 2
1 2 12 1 2
1
r r 2r r cos( ) 4 9 2.2.3cos 4 9 12cos 13 12. 7
66 3 2
? ? ?? pp p
+ - ?-? = +- - - == +- = - =
? ? ??
? ? ??
Illustration 3: The point whose abscissa is equal to its ordinate and which is equidistant from the points A(1, 0), 
B(0, 3) is   (JEE MAIN) 
Sol: Given, abscissa = ordinate. Therefore distance can be found by considering the co-ordinates of required point 
be P(k, k).
Now given PA = PB ? 
2 2
(k 1) k - + = 
2 2
k (k 3) + -
2k
2
 – 2k + 1 = 2k
2
 – 6k + 9 ? 4k = 8 ? k = 2   
4. SECTION FORMULA
Let R divide the two points P(x
1
, y
1
) and Q(x
2
, y
2
) internally in the ratio m:n. 
Let (x, y) be the co-ordinates of R.
Draw PM, QN, RK perpendicular to the x-axis. 
Also, draw PE and RF perpendicular to RK and QN.
Here, 
PR m
RQ n
= .
Triangles PRE and RFQ are similar. 
? 
PR PE
RQ RF
= ? 
PE m
RF n
=
But PE = x – x
1 
and RF = x
2
 – x.  
? 
1
2
xx m
x x n
-
=
-
 ? 
2 1
mx nx
x
m n
+
=
+
In the same way, 
ER m
FQ n
= 
i.e., 
1
2
y y m
y y n
-
=
-
? y = 
2 1
my ny
m n
+
+
The co-ordinates of R are 
2 1 2 1
mx nx my ny
,
m n m n
? ? + +
? ?
? ?
+ +
? ?
If R’ divides PQ externally, so that 
PR' m
QR' n
= , triangles PER’ and QR’F are similar.  
Figure 8.5
y
x’ O
y’
x
M N
K
P
Q
R’ F
E
? 
PR' PE
R'Q R'F
=
            
But PE = x – x
1 
and R’F = x – x
2
. 
? 
1
2
xx m
xx n
-
=
-
   i.e., x = 
2 1
mx nx
mn
-
-
Similarly, 
2 1
my ny
y
mn
-
=
-
.
The co-ordinates of R’ are 
2 1 2 1
mx nx my ny
,
mn mn
? ? - -
? ?
? ?
- -
? ?
 
y
x’
y’
x
P
R
Q
E
F
O
M N K
Figure 8.4
Alternate Method: 
PR' m m
R'Q n –n
= - =
By changing n into –n in the co-ordinates of R, we can obtain the co-ordinates 
of R’: 
2 1 2 1
mx nx my ny
,
mn mn
- -
- -
 
Cor. The mid-point joining the two points (x
1
, y
1
) and (x
2
, y
2
) is 
1 2 1 2
x xy y
,
2 2
? ? + +
? ?
? ?
? ? .
Cor. From the above cor., the co-ordinates of a point dividing PQ in the ratio ?:1 are 
1 2 1 2
x xy y
,
1 1
? ? + ? + ?
? ?
+ ? + ? ? ?
. Considering 
? as a variable parameter, i.e. of all values positive or negative, the co-ordinates of any point on the line joining the 
points (x
1
, y
1
) and (x
2
, y
2
) can be expressed in the above forms.
5. SPECIAL POINTS OF A TRIANGLE 
5.1 Centroid
Let the vertices of the triangle ABC be (x
1
, y
1
), (x
2
, y
2
) and (x
3
, y
3
), respectively.  
The mid-point D of BC is 
2 3 2 3
x xy y
,
2 2
? ? + +
? ?
? ?
? ?
G, the centroid, divides AD internally 
in the ratio 2:1. 
Let G be (x, y),
then x = 
( )
2 3 1
2. (x x )/2 1.x
21
+ +
+
 = 
1 2 3
x x x
3
+ +
 and
( )
2 3 1
2. (y y )/2 1.y
y
21
+ +
=
+
= 
1 2 3
y y y
3
++
? G is 
1 2 3 1 2 3
x x xy y y
,
3 3
? ? + + + +
? ?
? ?
? ?
.
5.2 Incentre
Let A (x
1
, y
1
), B (x
2
, y
2
), C (x
3
, y
3
) be the vertices of the triangle. 
Let AD bisect angle BAC and cut BC at D.
We know that 
BD AB c
DC AC b
= =
Hence the co-ordinates of D are 
3 2 3 2
cx bx cy by
,
c b c b
+ +
+ +
                      
A
B
C
D
I
Figure 8.7
Let (x, y) be the incentre of the triangle
CD b
BD c
= ? 
BC b c
DB c
+
= ? BD = 
ca
bc +
( )
AI AB c b c
ID BD a ca/(b c)
+
= = =
+
? 
( )
( )
3 2 1
1 2 3
3 2 1
1 2 3
(b c) (cx bx )/(c b) ax
ax bx cx
x ,
bc a a bc
(b c) (cy by )/(c b) ay
ay by cy
y
bc a a bc
+ + + +
+ +
= =
++ + +
+ + ++
+ +
= =
++ + +
( )
( )
3 2 1
1 2 3
3 2 1
1 2 3
(b c) (cx bx )/(c b) ax
ax bx cx
x ,
bc a a bc
(b c) (cy by )/(c b) ay
ay by cy
y
bc a a bc
+ + + +
+ +
= =
++ + +
+ + ++
+ +
= =
++ + +
Figure 8.6
C(x , y)
3 3
F
D
G
2
1
E (x , y)
1 1
A B(x , y)
2 2
5.3 Ex-centres
The centre of the circle which touches the side BC and the extended portions of sides         
Figure 8.8
I
3
I
2
I
1
A
B
C
AB and AC is called the ex-centre of ?ABC with respect to the vertex A. It is denoted by 
I
1
 and its co-ordinates are as follows:
I
1
 = 
1 2 3 1 2 3
ax bx cx ay by cy
,
ab c ab c
? ? - + + - + +
? ?
? ?
-+ + -+ +
? ?
 
Similarly ex-centres of ?ABC with respect to vertices B and C are denoted by I
2 
and I
3
, 
respectively, and
I
2
 = 
1 2 3 1 2 3
ax –bx cx ay by cy
,
a b c a b c
? ? + - +
? ?
? ?
-+ -+
? ?
,
I
3
 = 
1 2 3 1 2 3
ax bx cx ay by cy
,
ab c ab c
? ? + - + -
? ?
? ?
+ - + -
? ?
.
5.4 Circumcentre 
It is the point of intersection of perpendicular bisectors of the sides of the triangle.      
Figure 8.9
A(x , y)
1 1
B
O
E
C(x , y)
3 3
(x , y)
2 2
It is also the centre of a circle passing through the vertices of the triangle. If O is the 
circumcentre of any ?ABC, then, OA = OB = OC. 
Circumcentre:
1 2 3 1 2 3
x sin2A x sin2B x sin2C y sin2A y sin2B y sin2C
,
sin2A sin2A
? ? + + + +
? ?
? ?
S S
? ?
 
Note: For a right-angled triangle, its circumcentre is the mid-point of hypotenuse. 
A
B
C
Figure 8.10
5.5 Orthocentre
The point of intersection of altitudes of a triangle that can be obtained by solving the  
A(x , y)
1 1
(x , y)
2 2 (x , y)
3 3
B C
H
F
E
D
Figure 8.11
equation of any two altitudes is called Orthocentre. It is denoted by H 
Orthocentre: 
1 2 3 1 2 3
x tanA x tanB x tanC y tanA y tanB y tanC
,
tanA tanA
? ? + + + +
? ?
? ?
S S
? ?
Note: In a right angle triangle, orthocentre is the point where right angle is formed. 
Remarks:
(a) In an equilateral triangle, centroid, incentre, orthocentre, circumcentre coincide. 
(b) Orthocentre, centroid, and circumcentre are always collinear. Centroid divides the      
A
B C
D N
G
H
O
Figure 8.12
 
Orthocentre and circumcentre joining line in a 2: 1 ratio.
Proof: H, G and O are collinear and ?’s OGD & AGH are similar.
But OD (distance of c.c. from BC) = R cos A