Chapter 10
STRAIGHT LINES
vG 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
10.1 Introduction
We are familiar with two-dimensional 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 René Descartes
coordinate axes, coordinate plane, plotting of points in a (1596 -1650)
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 10.1.
We may note that the point (6, – 4) is at 6 units
distance from the y-axis measured along the positive
x-axis and at 4 units distance from the x-axis
measured along the negative y-axis. Similarly, the
point (3, 0) is at 3 units distance from the y-axis
measured along the positive x-axis and has zero
distance from the x-axis.
We also studied there following important Fig 10.1
formulae:
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,204 MATHEMATICS
I. Distance between the points P (x1, y1) and Q (x2, y2) is
(x – x1 ) + ( y2 – y1 )
2 2
PQ = 2
For example, distance between the points (6, – 4) and (3, 0) is
( 3 − 6 )2 + ( 0 + 4 )2 = 9 + 16 = 5 units.
II. The coordinates of a point dividing the line segment joining the points (x1, y1)
m x 2 + n x1 m y 2 + n y1
and (x2, y2) internally, in the ratio m: n are , .
m+n m + n
For example, the coordinates of the point which divides the line segment joining
1.( − 3) + 3.1
A (1, –3) and B (–3, 9) internally, in the ratio 1: 3 are given by x = =0
1+ 3
1.9 + 3. ( –3)
and y = = 0.
1+ 3
III. In particular, if m = n, the coordinates of the mid-point of the line segment
x1 + x2 y1 + y 2
joining the points (x1, y1) and (x2, y2) are , .
2 2
IV. Area of the triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3) is
1
x1 ( y 2 − y 3 ) + x 2 ( y 3 − y 1) + x 3 ( y1 − y 2 ) .
2
For example, the area of the triangle, whose vertices are (4, 4), (3, – 2) and (– 3, 16) is
1 − 54
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.
10.2 Slope of a Line
A line in a coordinate plane forms two angles with the x-axis, which are supplementary.
2020-21
, STRAIGHT LINES 205
The angle (say) θ made by the line l with positive
direction of x-axis and measured anti clockwise
is called the inclination of the line. Obviously
0° ≤ θ ≤ 180° (Fig 10.2).
We observe that lines parallel to x-axis, or
coinciding with x-axis, have inclination of 0°. The
inclination of a vertical line (parallel to or
coinciding with y-axis) is 90°.
Definition 1 If θ is the inclination of a line
l, then tan θ is called the slope or gradient of
the line l. Fig 10.2
The slope of a line whose inclination is 90° is not
defined.
The slope of a line is denoted by m.
Thus, m = tan θ, θ ≠ 90°
It may be observed that the slope of x-axis is zero and slope of y-axis is not defined.
10.2.1 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 non-vertical line l whose inclination
is θ. Obviously, x1 ≠ x2, otherwise the line
will become perpendicular to x-axis 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 x-axis and
PM perpendicular to RQ as shown in
Figs. 10.3 (i) and (ii).
Fig 10. 3 (i)
Case 1 When angle θ is acute:
In Fig 10.3 (i), ∠MPQ = θ. ... (1)
Therefore, slope of line l = m = tan θ.
MQ y2 − y1
But in ∆MPQ, we have tan θ = = . ... (2)
MP x2 − x1
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From equations (1) and (2), we have
y2 − y1
m= .
x2 − x1
Case II When angle θ is obtuse:
In Fig 10.3 (ii), we have
∠MPQ = 180° – θ.
Therefore, θ = 180° – ∠MPQ.
Now, slope of the line l
Fig 10. 3 (ii)
m = tan θ
= tan ( 180° – ∠MPQ) = – tan ∠MPQ
MQ y −y y2 − y1
= − =− 2 1 = .
MP x1 − x2 x2 − x1
Consequently, we see that in both the cases the slope m of the line through the points
y2 − y1
(x1, y1) and (x2, y2) is given by m = .
x2 − x1
10.2.2 Conditions for parallelism and perpendicularity of lines in terms of their
slopes In a coordinate plane, suppose that non-vertical lines l1 and l2 have slopes m1
and m2, respectively. Let their inclinations be α and
β, respectively.
If the line l1 is parallel to l2 (Fig 10.4), then their
inclinations are equal, i.e.,
α = β, and hence, tan α = tan β
Therefore m1 = m2, i.e., their slopes are equal.
Conversely, if the slope of two lines l1 and l2
is same, i.e.,
m1 = m2.
Fig 10. 4
Then tan α = tan β.
By the property of tangent function (between 0° and 180°), α = β.
Therefore, the lines are parallel.
2020-21
STRAIGHT LINES
vG 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
10.1 Introduction
We are familiar with two-dimensional 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 René Descartes
coordinate axes, coordinate plane, plotting of points in a (1596 -1650)
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 10.1.
We may note that the point (6, – 4) is at 6 units
distance from the y-axis measured along the positive
x-axis and at 4 units distance from the x-axis
measured along the negative y-axis. Similarly, the
point (3, 0) is at 3 units distance from the y-axis
measured along the positive x-axis and has zero
distance from the x-axis.
We also studied there following important Fig 10.1
formulae:
2020-21
,204 MATHEMATICS
I. Distance between the points P (x1, y1) and Q (x2, y2) is
(x – x1 ) + ( y2 – y1 )
2 2
PQ = 2
For example, distance between the points (6, – 4) and (3, 0) is
( 3 − 6 )2 + ( 0 + 4 )2 = 9 + 16 = 5 units.
II. The coordinates of a point dividing the line segment joining the points (x1, y1)
m x 2 + n x1 m y 2 + n y1
and (x2, y2) internally, in the ratio m: n are , .
m+n m + n
For example, the coordinates of the point which divides the line segment joining
1.( − 3) + 3.1
A (1, –3) and B (–3, 9) internally, in the ratio 1: 3 are given by x = =0
1+ 3
1.9 + 3. ( –3)
and y = = 0.
1+ 3
III. In particular, if m = n, the coordinates of the mid-point of the line segment
x1 + x2 y1 + y 2
joining the points (x1, y1) and (x2, y2) are , .
2 2
IV. Area of the triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3) is
1
x1 ( y 2 − y 3 ) + x 2 ( y 3 − y 1) + x 3 ( y1 − y 2 ) .
2
For example, the area of the triangle, whose vertices are (4, 4), (3, – 2) and (– 3, 16) is
1 − 54
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.
10.2 Slope of a Line
A line in a coordinate plane forms two angles with the x-axis, which are supplementary.
2020-21
, STRAIGHT LINES 205
The angle (say) θ made by the line l with positive
direction of x-axis and measured anti clockwise
is called the inclination of the line. Obviously
0° ≤ θ ≤ 180° (Fig 10.2).
We observe that lines parallel to x-axis, or
coinciding with x-axis, have inclination of 0°. The
inclination of a vertical line (parallel to or
coinciding with y-axis) is 90°.
Definition 1 If θ is the inclination of a line
l, then tan θ is called the slope or gradient of
the line l. Fig 10.2
The slope of a line whose inclination is 90° is not
defined.
The slope of a line is denoted by m.
Thus, m = tan θ, θ ≠ 90°
It may be observed that the slope of x-axis is zero and slope of y-axis is not defined.
10.2.1 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 non-vertical line l whose inclination
is θ. Obviously, x1 ≠ x2, otherwise the line
will become perpendicular to x-axis 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 x-axis and
PM perpendicular to RQ as shown in
Figs. 10.3 (i) and (ii).
Fig 10. 3 (i)
Case 1 When angle θ is acute:
In Fig 10.3 (i), ∠MPQ = θ. ... (1)
Therefore, slope of line l = m = tan θ.
MQ y2 − y1
But in ∆MPQ, we have tan θ = = . ... (2)
MP x2 − x1
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, 206 MATHEMATICS
From equations (1) and (2), we have
y2 − y1
m= .
x2 − x1
Case II When angle θ is obtuse:
In Fig 10.3 (ii), we have
∠MPQ = 180° – θ.
Therefore, θ = 180° – ∠MPQ.
Now, slope of the line l
Fig 10. 3 (ii)
m = tan θ
= tan ( 180° – ∠MPQ) = – tan ∠MPQ
MQ y −y y2 − y1
= − =− 2 1 = .
MP x1 − x2 x2 − x1
Consequently, we see that in both the cases the slope m of the line through the points
y2 − y1
(x1, y1) and (x2, y2) is given by m = .
x2 − x1
10.2.2 Conditions for parallelism and perpendicularity of lines in terms of their
slopes In a coordinate plane, suppose that non-vertical lines l1 and l2 have slopes m1
and m2, respectively. Let their inclinations be α and
β, respectively.
If the line l1 is parallel to l2 (Fig 10.4), then their
inclinations are equal, i.e.,
α = β, and hence, tan α = tan β
Therefore m1 = m2, i.e., their slopes are equal.
Conversely, if the slope of two lines l1 and l2
is same, i.e.,
m1 = m2.
Fig 10. 4
Then tan α = tan β.
By the property of tangent function (between 0° and 180°), α = β.
Therefore, the lines are parallel.
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