Page 1
vG G G G G eometry, as a logical system, is a means and even the most powerful
means to make children feel the strength of the human spirit that is
of their own spirit. – H. FREUDENTHALv
9.1 Introduction
W e are familiar with twodimensional coordinate geometry
from earlier classes. Mainly, it is a combination of algebra
and geometry. A systematic study of geometry by the use
of algebra was first carried out by celebrated French
philosopher and mathematician René Descartes, in his book
‘La Géométry, published in 1637. This book introduced the
notion of the equation of a curve and related analytical
methods into the study of geometry. The resulting
combination of analysis and geometry is referred now as
analytical geometry. In the earlier classes, we initiated
the study of coordinate geometry, where we studied about
coordinate axes, coordinate plane, plotting of points in a
plane, distance between two points, section formulae, etc. All these concepts are the
basics of coordinate geometry.
Let us have a brief recall of coordinate geometry done in earlier classes. To
recapitulate, the location of the points (6, – 4) and
(3, 0) in the XY plane is shown in Fig 9.1.
We may note that the point (6, – 4) is at 6 units
distance from the yaxis measured along the positive
xaxis and at 4 units distance from the xaxis
measured along the negative yaxis. Similarly, the
point (3, 0) is at 3 units distance from the yaxis
measured along the positive xaxis and has zero
distance from the xaxis.
We also studied there following important
formulae:
9 Chapter
STRAIGHT LINES
René Descartes
(1596 1650)
Fig 9.1
202425
Page 2
vG G G G G eometry, as a logical system, is a means and even the most powerful
means to make children feel the strength of the human spirit that is
of their own spirit. – H. FREUDENTHALv
9.1 Introduction
W e are familiar with twodimensional coordinate geometry
from earlier classes. Mainly, it is a combination of algebra
and geometry. A systematic study of geometry by the use
of algebra was first carried out by celebrated French
philosopher and mathematician René Descartes, in his book
‘La Géométry, published in 1637. This book introduced the
notion of the equation of a curve and related analytical
methods into the study of geometry. The resulting
combination of analysis and geometry is referred now as
analytical geometry. In the earlier classes, we initiated
the study of coordinate geometry, where we studied about
coordinate axes, coordinate plane, plotting of points in a
plane, distance between two points, section formulae, etc. All these concepts are the
basics of coordinate geometry.
Let us have a brief recall of coordinate geometry done in earlier classes. To
recapitulate, the location of the points (6, – 4) and
(3, 0) in the XY plane is shown in Fig 9.1.
We may note that the point (6, – 4) is at 6 units
distance from the yaxis measured along the positive
xaxis and at 4 units distance from the xaxis
measured along the negative yaxis. Similarly, the
point (3, 0) is at 3 units distance from the yaxis
measured along the positive xaxis and has zero
distance from the xaxis.
We also studied there following important
formulae:
9 Chapter
STRAIGHT LINES
René Descartes
(1596 1650)
Fig 9.1
202425
152 MATHEMATICS
I. Distance between the points P (x
1,
y
1
) and Q (x
2
, y
2
) is
( ) ( ) 1
2
2
2 2 1
PQ x – x y – y = +
For example, distance between the points (6, – 4) and (3, 0) is
( ) ( )
2 2
3 6 0 4 9 16 5  + + = + = units.
II. The coordinates of a point dividing the line segment joining the points (x
1,
y
1
)
and (x
2
, y
2
) internally, in the ratio m: n are
?
?
?
?
?
?
?
?
+
+
+
+
n m
y n y m
n m
x
n
x
m
1 2 1 2
, .
For example, the coordinates of the point which divides the line segment joining
A (1, –3) and B (–3, 9) internally, in the ratio 1: 3 are given by
1 ( 3) 3 1
0
1 3
. .
x
 +
= =
+
and
( ) 1.9 + 3. –3
= = 0.
1 + 3
y
III. In particular, if m = n, the coordinates of the midpoint of the line segment
joining the points (x
1,
y
1
) and (x
2
, y
2
) are ?
?
?
?
?
? + +
2
,
2
2 1 2 1
y y
x x
.
IV . Area of the triangle whose vertices are (x
1,
y
1
), (x
2
, y
2
) and (x
3
, y
3
) is
( ) ( ) ( ) 1 2 3
2 3 3 1 1 2
1
2
 +  +  y y y y y y
x x x .
For example, the area of the triangle, whose vertices are (4, 4), (3, – 2) and (– 3, 16) is
54
1
4( 2 16) 3(16 4) ( 3)(4 2) 27.
2 2

  +  +  + = =
Remark If the area of the triangle ABC is zero, then three points A, B and C lie on
a line, i.e., they are collinear.
In the this Chapter, we shall continue the study of coordinate geometry to study
properties of the simplest geometric figure – straight line. Despite its simplicity, the
line is a vital concept of geometry and enters into our daily experiences in numerous
interesting and useful ways. Main focus is on representing the line algebraically, for
which slope is most essential.
9.2 Slope of a Line
A line in a coordinate plane forms two angles with the xaxis, which are supplementary.
202425
Page 3
vG G G G G eometry, as a logical system, is a means and even the most powerful
means to make children feel the strength of the human spirit that is
of their own spirit. – H. FREUDENTHALv
9.1 Introduction
W e are familiar with twodimensional coordinate geometry
from earlier classes. Mainly, it is a combination of algebra
and geometry. A systematic study of geometry by the use
of algebra was first carried out by celebrated French
philosopher and mathematician René Descartes, in his book
‘La Géométry, published in 1637. This book introduced the
notion of the equation of a curve and related analytical
methods into the study of geometry. The resulting
combination of analysis and geometry is referred now as
analytical geometry. In the earlier classes, we initiated
the study of coordinate geometry, where we studied about
coordinate axes, coordinate plane, plotting of points in a
plane, distance between two points, section formulae, etc. All these concepts are the
basics of coordinate geometry.
Let us have a brief recall of coordinate geometry done in earlier classes. To
recapitulate, the location of the points (6, – 4) and
(3, 0) in the XY plane is shown in Fig 9.1.
We may note that the point (6, – 4) is at 6 units
distance from the yaxis measured along the positive
xaxis and at 4 units distance from the xaxis
measured along the negative yaxis. Similarly, the
point (3, 0) is at 3 units distance from the yaxis
measured along the positive xaxis and has zero
distance from the xaxis.
We also studied there following important
formulae:
9 Chapter
STRAIGHT LINES
René Descartes
(1596 1650)
Fig 9.1
202425
152 MATHEMATICS
I. Distance between the points P (x
1,
y
1
) and Q (x
2
, y
2
) is
( ) ( ) 1
2
2
2 2 1
PQ x – x y – y = +
For example, distance between the points (6, – 4) and (3, 0) is
( ) ( )
2 2
3 6 0 4 9 16 5  + + = + = units.
II. The coordinates of a point dividing the line segment joining the points (x
1,
y
1
)
and (x
2
, y
2
) internally, in the ratio m: n are
?
?
?
?
?
?
?
?
+
+
+
+
n m
y n y m
n m
x
n
x
m
1 2 1 2
, .
For example, the coordinates of the point which divides the line segment joining
A (1, –3) and B (–3, 9) internally, in the ratio 1: 3 are given by
1 ( 3) 3 1
0
1 3
. .
x
 +
= =
+
and
( ) 1.9 + 3. –3
= = 0.
1 + 3
y
III. In particular, if m = n, the coordinates of the midpoint of the line segment
joining the points (x
1,
y
1
) and (x
2
, y
2
) are ?
?
?
?
?
? + +
2
,
2
2 1 2 1
y y
x x
.
IV . Area of the triangle whose vertices are (x
1,
y
1
), (x
2
, y
2
) and (x
3
, y
3
) is
( ) ( ) ( ) 1 2 3
2 3 3 1 1 2
1
2
 +  +  y y y y y y
x x x .
For example, the area of the triangle, whose vertices are (4, 4), (3, – 2) and (– 3, 16) is
54
1
4( 2 16) 3(16 4) ( 3)(4 2) 27.
2 2

  +  +  + = =
Remark If the area of the triangle ABC is zero, then three points A, B and C lie on
a line, i.e., they are collinear.
In the this Chapter, we shall continue the study of coordinate geometry to study
properties of the simplest geometric figure – straight line. Despite its simplicity, the
line is a vital concept of geometry and enters into our daily experiences in numerous
interesting and useful ways. Main focus is on representing the line algebraically, for
which slope is most essential.
9.2 Slope of a Line
A line in a coordinate plane forms two angles with the xaxis, which are supplementary.
202425
STRAIGHT LINES 153
The angle (say) ? made by the line l with positive
direction of xaxis and measured anti clockwise
is called the inclination of the line. Obviously
0° = ? = 180° (Fig 9.2).
We observe that lines parallel to xaxis, or
coinciding with xaxis, have inclination of 0°. The
inclination of a vertical line (parallel to or
coinciding with yaxis) is 90°.
Definition 1 If ? is the inclination of a line
l, then tan ? is called the slope or gradient of
the line l.
The slope of a line whose inclination is 90° is not
defined.
The slope of a line is denoted by m.
Thus, m = tan ?, ? ? 90°
It may be observed that the slope of xaxis is zero and slope of yaxis is not defined.
9.2.1 Slope of a line when coordinates of any two points on the line are given
We know that a line is completely determined when we are given two points on it.
Hence, we proceed to find the slope of a
line in terms of the coordinates of two points
on the line.
Let P(x
1
, y
1
) and Q(x
2
, y
2
) be two
points on nonvertical line l whose inclination
is ?. Obviously, x
1
? x
2
, otherwise the line
will become perpendicular to xaxis and its
slope will not be defined. The inclination of
the line l may be acute or obtuse. Let us
take these two cases.
Draw perpendicular QR to xaxis and
PM perpendicular to RQ as shown in
Figs. 9.3 (i) and (ii).
Case 1 When angle ? is acute:
In Fig 9.3 (i), ?MPQ = ?. ... (1)
Therefore, slope of line l = m = tan ?.
But in ?MPQ, we have
2 1
2 1
MQ
tan? .
MP
y y
x x
 = =
 ... (2)
Fig 9.2
Fig 9. 3 (i)
202425
Page 4
vG G G G G eometry, as a logical system, is a means and even the most powerful
means to make children feel the strength of the human spirit that is
of their own spirit. – H. FREUDENTHALv
9.1 Introduction
W e are familiar with twodimensional coordinate geometry
from earlier classes. Mainly, it is a combination of algebra
and geometry. A systematic study of geometry by the use
of algebra was first carried out by celebrated French
philosopher and mathematician René Descartes, in his book
‘La Géométry, published in 1637. This book introduced the
notion of the equation of a curve and related analytical
methods into the study of geometry. The resulting
combination of analysis and geometry is referred now as
analytical geometry. In the earlier classes, we initiated
the study of coordinate geometry, where we studied about
coordinate axes, coordinate plane, plotting of points in a
plane, distance between two points, section formulae, etc. All these concepts are the
basics of coordinate geometry.
Let us have a brief recall of coordinate geometry done in earlier classes. To
recapitulate, the location of the points (6, – 4) and
(3, 0) in the XY plane is shown in Fig 9.1.
We may note that the point (6, – 4) is at 6 units
distance from the yaxis measured along the positive
xaxis and at 4 units distance from the xaxis
measured along the negative yaxis. Similarly, the
point (3, 0) is at 3 units distance from the yaxis
measured along the positive xaxis and has zero
distance from the xaxis.
We also studied there following important
formulae:
9 Chapter
STRAIGHT LINES
René Descartes
(1596 1650)
Fig 9.1
202425
152 MATHEMATICS
I. Distance between the points P (x
1,
y
1
) and Q (x
2
, y
2
) is
( ) ( ) 1
2
2
2 2 1
PQ x – x y – y = +
For example, distance between the points (6, – 4) and (3, 0) is
( ) ( )
2 2
3 6 0 4 9 16 5  + + = + = units.
II. The coordinates of a point dividing the line segment joining the points (x
1,
y
1
)
and (x
2
, y
2
) internally, in the ratio m: n are
?
?
?
?
?
?
?
?
+
+
+
+
n m
y n y m
n m
x
n
x
m
1 2 1 2
, .
For example, the coordinates of the point which divides the line segment joining
A (1, –3) and B (–3, 9) internally, in the ratio 1: 3 are given by
1 ( 3) 3 1
0
1 3
. .
x
 +
= =
+
and
( ) 1.9 + 3. –3
= = 0.
1 + 3
y
III. In particular, if m = n, the coordinates of the midpoint of the line segment
joining the points (x
1,
y
1
) and (x
2
, y
2
) are ?
?
?
?
?
? + +
2
,
2
2 1 2 1
y y
x x
.
IV . Area of the triangle whose vertices are (x
1,
y
1
), (x
2
, y
2
) and (x
3
, y
3
) is
( ) ( ) ( ) 1 2 3
2 3 3 1 1 2
1
2
 +  +  y y y y y y
x x x .
For example, the area of the triangle, whose vertices are (4, 4), (3, – 2) and (– 3, 16) is
54
1
4( 2 16) 3(16 4) ( 3)(4 2) 27.
2 2

  +  +  + = =
Remark If the area of the triangle ABC is zero, then three points A, B and C lie on
a line, i.e., they are collinear.
In the this Chapter, we shall continue the study of coordinate geometry to study
properties of the simplest geometric figure – straight line. Despite its simplicity, the
line is a vital concept of geometry and enters into our daily experiences in numerous
interesting and useful ways. Main focus is on representing the line algebraically, for
which slope is most essential.
9.2 Slope of a Line
A line in a coordinate plane forms two angles with the xaxis, which are supplementary.
202425
STRAIGHT LINES 153
The angle (say) ? made by the line l with positive
direction of xaxis and measured anti clockwise
is called the inclination of the line. Obviously
0° = ? = 180° (Fig 9.2).
We observe that lines parallel to xaxis, or
coinciding with xaxis, have inclination of 0°. The
inclination of a vertical line (parallel to or
coinciding with yaxis) is 90°.
Definition 1 If ? is the inclination of a line
l, then tan ? is called the slope or gradient of
the line l.
The slope of a line whose inclination is 90° is not
defined.
The slope of a line is denoted by m.
Thus, m = tan ?, ? ? 90°
It may be observed that the slope of xaxis is zero and slope of yaxis is not defined.
9.2.1 Slope of a line when coordinates of any two points on the line are given
We know that a line is completely determined when we are given two points on it.
Hence, we proceed to find the slope of a
line in terms of the coordinates of two points
on the line.
Let P(x
1
, y
1
) and Q(x
2
, y
2
) be two
points on nonvertical line l whose inclination
is ?. Obviously, x
1
? x
2
, otherwise the line
will become perpendicular to xaxis and its
slope will not be defined. The inclination of
the line l may be acute or obtuse. Let us
take these two cases.
Draw perpendicular QR to xaxis and
PM perpendicular to RQ as shown in
Figs. 9.3 (i) and (ii).
Case 1 When angle ? is acute:
In Fig 9.3 (i), ?MPQ = ?. ... (1)
Therefore, slope of line l = m = tan ?.
But in ?MPQ, we have
2 1
2 1
MQ
tan? .
MP
y y
x x
 = =
 ... (2)
Fig 9.2
Fig 9. 3 (i)
202425
154 MATHEMATICS
From equations (1) and (2), we have
2 1
2 1
.
y y
m
x x
 =
 Case II When angle ? is obtuse:
In Fig 9.3 (ii), we have
?MPQ = 180° – ?.
Therefore, ? = 180° – ?MPQ.
Now, slope of the line l
m = tan ?
= tan ( 180° – ?MPQ) = – tan ?MPQ
=
2 1
1 2
MQ
MP
y y
x x

 = 

=
2 1
2 1
y y
.
x x


Consequently, we see that in both the cases the slope m of the line through the points
(x
1
, y
1
) and (x
2
, y
2
) is given by
2 1
2 1
y y
m
x x
 =
 .
9.2.2 Conditions for parallelism and perpendicularity of lines in terms of their
slopes In a coordinate plane, suppose that nonvertical lines l
1
and l
2
have slopes m
1
and m
2
, respectively. Let their inclinations be a and
ß, respectively.
If the line l
1
is parallel to l
2
(Fig 9.4), then their
inclinations are equal, i.e.,
a = ß, and hence, tan a = tan ß
Therefore m
1
= m
2
, i.e., their slopes are equal.
Conversely, if the slope of two lines l
1
and l
2
is same, i.e.,
m
1
= m
2
.
Then tan a = tan ß.
By the property of tangent function (between 0° and 180°), a = ß.
Therefore, the lines are parallel.
Fig 9. 3 (ii)
Fig 9. 4
202425
Page 5
vG G G G G eometry, as a logical system, is a means and even the most powerful
means to make children feel the strength of the human spirit that is
of their own spirit. – H. FREUDENTHALv
9.1 Introduction
W e are familiar with twodimensional coordinate geometry
from earlier classes. Mainly, it is a combination of algebra
and geometry. A systematic study of geometry by the use
of algebra was first carried out by celebrated French
philosopher and mathematician René Descartes, in his book
‘La Géométry, published in 1637. This book introduced the
notion of the equation of a curve and related analytical
methods into the study of geometry. The resulting
combination of analysis and geometry is referred now as
analytical geometry. In the earlier classes, we initiated
the study of coordinate geometry, where we studied about
coordinate axes, coordinate plane, plotting of points in a
plane, distance between two points, section formulae, etc. All these concepts are the
basics of coordinate geometry.
Let us have a brief recall of coordinate geometry done in earlier classes. To
recapitulate, the location of the points (6, – 4) and
(3, 0) in the XY plane is shown in Fig 9.1.
We may note that the point (6, – 4) is at 6 units
distance from the yaxis measured along the positive
xaxis and at 4 units distance from the xaxis
measured along the negative yaxis. Similarly, the
point (3, 0) is at 3 units distance from the yaxis
measured along the positive xaxis and has zero
distance from the xaxis.
We also studied there following important
formulae:
9 Chapter
STRAIGHT LINES
René Descartes
(1596 1650)
Fig 9.1
202425
152 MATHEMATICS
I. Distance between the points P (x
1,
y
1
) and Q (x
2
, y
2
) is
( ) ( ) 1
2
2
2 2 1
PQ x – x y – y = +
For example, distance between the points (6, – 4) and (3, 0) is
( ) ( )
2 2
3 6 0 4 9 16 5  + + = + = units.
II. The coordinates of a point dividing the line segment joining the points (x
1,
y
1
)
and (x
2
, y
2
) internally, in the ratio m: n are
?
?
?
?
?
?
?
?
+
+
+
+
n m
y n y m
n m
x
n
x
m
1 2 1 2
, .
For example, the coordinates of the point which divides the line segment joining
A (1, –3) and B (–3, 9) internally, in the ratio 1: 3 are given by
1 ( 3) 3 1
0
1 3
. .
x
 +
= =
+
and
( ) 1.9 + 3. –3
= = 0.
1 + 3
y
III. In particular, if m = n, the coordinates of the midpoint of the line segment
joining the points (x
1,
y
1
) and (x
2
, y
2
) are ?
?
?
?
?
? + +
2
,
2
2 1 2 1
y y
x x
.
IV . Area of the triangle whose vertices are (x
1,
y
1
), (x
2
, y
2
) and (x
3
, y
3
) is
( ) ( ) ( ) 1 2 3
2 3 3 1 1 2
1
2
 +  +  y y y y y y
x x x .
For example, the area of the triangle, whose vertices are (4, 4), (3, – 2) and (– 3, 16) is
54
1
4( 2 16) 3(16 4) ( 3)(4 2) 27.
2 2

  +  +  + = =
Remark If the area of the triangle ABC is zero, then three points A, B and C lie on
a line, i.e., they are collinear.
In the this Chapter, we shall continue the study of coordinate geometry to study
properties of the simplest geometric figure – straight line. Despite its simplicity, the
line is a vital concept of geometry and enters into our daily experiences in numerous
interesting and useful ways. Main focus is on representing the line algebraically, for
which slope is most essential.
9.2 Slope of a Line
A line in a coordinate plane forms two angles with the xaxis, which are supplementary.
202425
STRAIGHT LINES 153
The angle (say) ? made by the line l with positive
direction of xaxis and measured anti clockwise
is called the inclination of the line. Obviously
0° = ? = 180° (Fig 9.2).
We observe that lines parallel to xaxis, or
coinciding with xaxis, have inclination of 0°. The
inclination of a vertical line (parallel to or
coinciding with yaxis) is 90°.
Definition 1 If ? is the inclination of a line
l, then tan ? is called the slope or gradient of
the line l.
The slope of a line whose inclination is 90° is not
defined.
The slope of a line is denoted by m.
Thus, m = tan ?, ? ? 90°
It may be observed that the slope of xaxis is zero and slope of yaxis is not defined.
9.2.1 Slope of a line when coordinates of any two points on the line are given
We know that a line is completely determined when we are given two points on it.
Hence, we proceed to find the slope of a
line in terms of the coordinates of two points
on the line.
Let P(x
1
, y
1
) and Q(x
2
, y
2
) be two
points on nonvertical line l whose inclination
is ?. Obviously, x
1
? x
2
, otherwise the line
will become perpendicular to xaxis and its
slope will not be defined. The inclination of
the line l may be acute or obtuse. Let us
take these two cases.
Draw perpendicular QR to xaxis and
PM perpendicular to RQ as shown in
Figs. 9.3 (i) and (ii).
Case 1 When angle ? is acute:
In Fig 9.3 (i), ?MPQ = ?. ... (1)
Therefore, slope of line l = m = tan ?.
But in ?MPQ, we have
2 1
2 1
MQ
tan? .
MP
y y
x x
 = =
 ... (2)
Fig 9.2
Fig 9. 3 (i)
202425
154 MATHEMATICS
From equations (1) and (2), we have
2 1
2 1
.
y y
m
x x
 =
 Case II When angle ? is obtuse:
In Fig 9.3 (ii), we have
?MPQ = 180° – ?.
Therefore, ? = 180° – ?MPQ.
Now, slope of the line l
m = tan ?
= tan ( 180° – ?MPQ) = – tan ?MPQ
=
2 1
1 2
MQ
MP
y y
x x

 = 

=
2 1
2 1
y y
.
x x


Consequently, we see that in both the cases the slope m of the line through the points
(x
1
, y
1
) and (x
2
, y
2
) is given by
2 1
2 1
y y
m
x x
 =
 .
9.2.2 Conditions for parallelism and perpendicularity of lines in terms of their
slopes In a coordinate plane, suppose that nonvertical lines l
1
and l
2
have slopes m
1
and m
2
, respectively. Let their inclinations be a and
ß, respectively.
If the line l
1
is parallel to l
2
(Fig 9.4), then their
inclinations are equal, i.e.,
a = ß, and hence, tan a = tan ß
Therefore m
1
= m
2
, i.e., their slopes are equal.
Conversely, if the slope of two lines l
1
and l
2
is same, i.e.,
m
1
= m
2
.
Then tan a = tan ß.
By the property of tangent function (between 0° and 180°), a = ß.
Therefore, the lines are parallel.
Fig 9. 3 (ii)
Fig 9. 4
202425
STRAIGHT LINES 155
Hence, two non vertical lines l
1
and l
2
are parallel if and only if their slopes
are equal.
If the lines l
1
and l
2
are perpendicular (Fig 9.5), then ß = a + 90°.
Therefore, tan ß = tan (a + 90°)
= – cot a =
1
tana
 i.e., m
2
=
1
1
m
 or m
1
m
2
= – 1
Conversely, if m
1
m
2
= – 1, i.e., tan a tan ß = – 1.
Then tan a = – cot ß = tan (ß + 90°) or tan (ß – 90°)
Therefore, a and ß differ by 90°.
Thus, lines l
1
and l
2
are perpendicular to each other.
Hence, two nonvertical lines are perpendicular to each other if and only if
their slopes are negative reciprocals of each other,
i.e., m
2
=
1
1
m
 or, m
1
m
2
= – 1.
Let us consider the following example.
Example 1 Find the slope of the lines:
(a) Passing through the points (3, – 2) and (–1, 4),
(b) Passing through the points (3, – 2) and (7, – 2),
(c) Passing through the points (3, – 2) and (3, 4),
(d) Making inclination of 60° with the positive direction of xaxis.
Solution (a) The slope of the line through (3, – 2) and (– 1, 4) is
4 ( 2) 6 3
1 3 4 2
m
 
= = = 
  
.
(b) The slope of the line through the points (3, – 2) and (7, – 2) is
–2 – (–2) 0
= = = 0
7 – 3 4
m
.
(c) The slope of the line through the points (3, – 2) and (3, 4) is
Fig 9. 5
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