Page 1
Straight Line
The knowledge of which geometry aims is the knowledge of the eternal...... Plato
A french mathematician and a greatest philosopher named Rene Descartes, pioneered the use of
algebra in Geometry. He suggested methods to study geometry by algebraic methods without making
direct reference to the actual figures
This geometry was called co-ordinate geometry or analytical geometry and it is the branch of geometry
in which algebraic equations are used to denote points, lines and curves.
Rectangular cartesian co-ordinate systems:
We shall right now focus on two-dimensional co-ordinate geometry in which two perpendicular lines
called co-ordinate axes (x-axis and ?? -axis) are used to locate a point in the plane.
?? is called origin. Any point ?? in this plane can be represented by a unique ordered pair ( ?? , ?? ), which
are called co-ordinates of that point. ?? is called ?? co-ordinate or abscissa and ?? is called ?? co-ordinate or
ordinate. The two perpendicular lines xox' and yoy' divide the plane in four regions which are called
quadrants, numbered as shown in the figure.
Let us look at some of the formulae linked with points now.
Distance Formula:
In rectangular Cartesian coordinate system
Page 2
Straight Line
The knowledge of which geometry aims is the knowledge of the eternal...... Plato
A french mathematician and a greatest philosopher named Rene Descartes, pioneered the use of
algebra in Geometry. He suggested methods to study geometry by algebraic methods without making
direct reference to the actual figures
This geometry was called co-ordinate geometry or analytical geometry and it is the branch of geometry
in which algebraic equations are used to denote points, lines and curves.
Rectangular cartesian co-ordinate systems:
We shall right now focus on two-dimensional co-ordinate geometry in which two perpendicular lines
called co-ordinate axes (x-axis and ?? -axis) are used to locate a point in the plane.
?? is called origin. Any point ?? in this plane can be represented by a unique ordered pair ( ?? , ?? ), which
are called co-ordinates of that point. ?? is called ?? co-ordinate or abscissa and ?? is called ?? co-ordinate or
ordinate. The two perpendicular lines xox' and yoy' divide the plane in four regions which are called
quadrants, numbered as shown in the figure.
Let us look at some of the formulae linked with points now.
Distance Formula:
In rectangular Cartesian coordinate system
The distance between the points ?? (?? 1
, ?? 1
) and ?? (?? 2
, ?? 2
) is = v(?? 1
- ?? 2
)
2
+ (?? 1
- ?? 2
)
2
.
Problem1: Find the value of ?? , if the distance between the points (?? , 8) and (4,3) is 13
Solution: Let ?? (?? , 8) and ?? (4,3) be the given points. Then ???? = 13 (given)
v(?? - 4)
2
+ (8 - 3)
2
= 13 ? (?? - 4)
2
+ 25 = 169 ? ?? = 16 ???? ?? = -8
Section Formula:
If ?? (?? , ?? ) divides the line joining ?? (?? 1
, ?? 1
) & ?? (?? 2
, ?? 2
) in the ratio ?? : ?? , then;
?? =
?? ?? 2
+ ?? ?? 1
?? + ?? ; ?? =
?? ?? 2
+ ?? ?? 1
?? + ??
Notes: (i) If
?? ?? is positive, the division is internal, but if
?? ?? is negative, the division is external.
(ii) If ?? divides ???? internally in the ratio ?? : ?? &?? divides ???? externally in the ratio ?? : ?? then ?? &?? are
said to be harmonic conjugate of each other w.r.t. AB.
Mathematically,
2
????
=
1
????
+
1
????
i.e. ???? , ???? &???? are in H.P.
Problem 2: Find the co-ordinates of the point which divides the line segment joining the points (2,5)
and (-3,7) in the ratio 2: 3 (i) internally and (ii) externally.
Solution: Let ?? (?? , ?? ) be the required point.
(i) For internal division:
(?? , ?? ) = (
-6 + 6
2 + 3
,
14 + 15
2 + 3
) = (0,
29
5
)
(ii) For external division
(?? , ?? ) = (
-6 - 6
2 - 3
,
14 - 15
2 - 3
) = (12,1)
Page 3
Straight Line
The knowledge of which geometry aims is the knowledge of the eternal...... Plato
A french mathematician and a greatest philosopher named Rene Descartes, pioneered the use of
algebra in Geometry. He suggested methods to study geometry by algebraic methods without making
direct reference to the actual figures
This geometry was called co-ordinate geometry or analytical geometry and it is the branch of geometry
in which algebraic equations are used to denote points, lines and curves.
Rectangular cartesian co-ordinate systems:
We shall right now focus on two-dimensional co-ordinate geometry in which two perpendicular lines
called co-ordinate axes (x-axis and ?? -axis) are used to locate a point in the plane.
?? is called origin. Any point ?? in this plane can be represented by a unique ordered pair ( ?? , ?? ), which
are called co-ordinates of that point. ?? is called ?? co-ordinate or abscissa and ?? is called ?? co-ordinate or
ordinate. The two perpendicular lines xox' and yoy' divide the plane in four regions which are called
quadrants, numbered as shown in the figure.
Let us look at some of the formulae linked with points now.
Distance Formula:
In rectangular Cartesian coordinate system
The distance between the points ?? (?? 1
, ?? 1
) and ?? (?? 2
, ?? 2
) is = v(?? 1
- ?? 2
)
2
+ (?? 1
- ?? 2
)
2
.
Problem1: Find the value of ?? , if the distance between the points (?? , 8) and (4,3) is 13
Solution: Let ?? (?? , 8) and ?? (4,3) be the given points. Then ???? = 13 (given)
v(?? - 4)
2
+ (8 - 3)
2
= 13 ? (?? - 4)
2
+ 25 = 169 ? ?? = 16 ???? ?? = -8
Section Formula:
If ?? (?? , ?? ) divides the line joining ?? (?? 1
, ?? 1
) & ?? (?? 2
, ?? 2
) in the ratio ?? : ?? , then;
?? =
?? ?? 2
+ ?? ?? 1
?? + ?? ; ?? =
?? ?? 2
+ ?? ?? 1
?? + ??
Notes: (i) If
?? ?? is positive, the division is internal, but if
?? ?? is negative, the division is external.
(ii) If ?? divides ???? internally in the ratio ?? : ?? &?? divides ???? externally in the ratio ?? : ?? then ?? &?? are
said to be harmonic conjugate of each other w.r.t. AB.
Mathematically,
2
????
=
1
????
+
1
????
i.e. ???? , ???? &???? are in H.P.
Problem 2: Find the co-ordinates of the point which divides the line segment joining the points (2,5)
and (-3,7) in the ratio 2: 3 (i) internally and (ii) externally.
Solution: Let ?? (?? , ?? ) be the required point.
(i) For internal division:
(?? , ?? ) = (
-6 + 6
2 + 3
,
14 + 15
2 + 3
) = (0,
29
5
)
(ii) For external division
(?? , ?? ) = (
-6 - 6
2 - 3
,
14 - 15
2 - 3
) = (12,1)
Problem 3: Find the co-ordinates of points which trisect the line segment joining (2, -3) and (4,5).
Solution: Let ?? (2, -3) and ?? (4,5) be the given points. Let the points of trisection be ?? and ?? . Then ???? =
???? = ???? = ?? (say)
? ???? = ???? + ???? = 2?? and ???? = ???? + ???? = 2??
? ???? : ???? = ?? : 2?? = 1: 2 and ???? : ???? = 2?? : ?? = 2: 1
So ?? divides ???? internally in the ratio 1: 2 while ?? divides internally in the ratio 2: 1
So ?? divides ???? internally in the ratio 1: 2 while ?? divides internally in the ratio 2: 1
? the co-ordinates of ?? are (
4+4
1+2
,
5-6
1+2
) = (
8
3
, -
1
3
)
and the co-ordinates of ?? are (
8+2
1+2
,
10-3
1+2
) = (
10
3
,
7
3
)
Hence, the points of trisection are (
8
3
, -
1
3
) and (
10
3
,
7
3
)
The ratio in which a given line divides the line segment
joining two points:
Let the given line ???? + ???? + ?? = 0 divide the line segment joining ?? (?? 1
, ?? 1
)&?? (?? 2
, ?? 2
) in the ratio ?? : ?? ,
then
?? ?? = -
?? ?? 1
+?? ?? 1
+?? ?? ?? 2
+?? ?? 2
+?? . If ?? &?? are on the same side of the given line then ?? /?? is negative but if ?? & ?? are
on opposite sides of the given line, then ?? /?? is positive
Example# 4: Find the ratio in which the line joining the points ?? (1,2) and ?? (-3,4) is divided by the line
?? + ?? - 5 = 0.
Solution: Let the line ?? + ?? = 5 divides ???? in the ratio ?? : 1 at ??
Page 4
Straight Line
The knowledge of which geometry aims is the knowledge of the eternal...... Plato
A french mathematician and a greatest philosopher named Rene Descartes, pioneered the use of
algebra in Geometry. He suggested methods to study geometry by algebraic methods without making
direct reference to the actual figures
This geometry was called co-ordinate geometry or analytical geometry and it is the branch of geometry
in which algebraic equations are used to denote points, lines and curves.
Rectangular cartesian co-ordinate systems:
We shall right now focus on two-dimensional co-ordinate geometry in which two perpendicular lines
called co-ordinate axes (x-axis and ?? -axis) are used to locate a point in the plane.
?? is called origin. Any point ?? in this plane can be represented by a unique ordered pair ( ?? , ?? ), which
are called co-ordinates of that point. ?? is called ?? co-ordinate or abscissa and ?? is called ?? co-ordinate or
ordinate. The two perpendicular lines xox' and yoy' divide the plane in four regions which are called
quadrants, numbered as shown in the figure.
Let us look at some of the formulae linked with points now.
Distance Formula:
In rectangular Cartesian coordinate system
The distance between the points ?? (?? 1
, ?? 1
) and ?? (?? 2
, ?? 2
) is = v(?? 1
- ?? 2
)
2
+ (?? 1
- ?? 2
)
2
.
Problem1: Find the value of ?? , if the distance between the points (?? , 8) and (4,3) is 13
Solution: Let ?? (?? , 8) and ?? (4,3) be the given points. Then ???? = 13 (given)
v(?? - 4)
2
+ (8 - 3)
2
= 13 ? (?? - 4)
2
+ 25 = 169 ? ?? = 16 ???? ?? = -8
Section Formula:
If ?? (?? , ?? ) divides the line joining ?? (?? 1
, ?? 1
) & ?? (?? 2
, ?? 2
) in the ratio ?? : ?? , then;
?? =
?? ?? 2
+ ?? ?? 1
?? + ?? ; ?? =
?? ?? 2
+ ?? ?? 1
?? + ??
Notes: (i) If
?? ?? is positive, the division is internal, but if
?? ?? is negative, the division is external.
(ii) If ?? divides ???? internally in the ratio ?? : ?? &?? divides ???? externally in the ratio ?? : ?? then ?? &?? are
said to be harmonic conjugate of each other w.r.t. AB.
Mathematically,
2
????
=
1
????
+
1
????
i.e. ???? , ???? &???? are in H.P.
Problem 2: Find the co-ordinates of the point which divides the line segment joining the points (2,5)
and (-3,7) in the ratio 2: 3 (i) internally and (ii) externally.
Solution: Let ?? (?? , ?? ) be the required point.
(i) For internal division:
(?? , ?? ) = (
-6 + 6
2 + 3
,
14 + 15
2 + 3
) = (0,
29
5
)
(ii) For external division
(?? , ?? ) = (
-6 - 6
2 - 3
,
14 - 15
2 - 3
) = (12,1)
Problem 3: Find the co-ordinates of points which trisect the line segment joining (2, -3) and (4,5).
Solution: Let ?? (2, -3) and ?? (4,5) be the given points. Let the points of trisection be ?? and ?? . Then ???? =
???? = ???? = ?? (say)
? ???? = ???? + ???? = 2?? and ???? = ???? + ???? = 2??
? ???? : ???? = ?? : 2?? = 1: 2 and ???? : ???? = 2?? : ?? = 2: 1
So ?? divides ???? internally in the ratio 1: 2 while ?? divides internally in the ratio 2: 1
So ?? divides ???? internally in the ratio 1: 2 while ?? divides internally in the ratio 2: 1
? the co-ordinates of ?? are (
4+4
1+2
,
5-6
1+2
) = (
8
3
, -
1
3
)
and the co-ordinates of ?? are (
8+2
1+2
,
10-3
1+2
) = (
10
3
,
7
3
)
Hence, the points of trisection are (
8
3
, -
1
3
) and (
10
3
,
7
3
)
The ratio in which a given line divides the line segment
joining two points:
Let the given line ???? + ???? + ?? = 0 divide the line segment joining ?? (?? 1
, ?? 1
)&?? (?? 2
, ?? 2
) in the ratio ?? : ?? ,
then
?? ?? = -
?? ?? 1
+?? ?? 1
+?? ?? ?? 2
+?? ?? 2
+?? . If ?? &?? are on the same side of the given line then ?? /?? is negative but if ?? & ?? are
on opposite sides of the given line, then ?? /?? is positive
Example# 4: Find the ratio in which the line joining the points ?? (1,2) and ?? (-3,4) is divided by the line
?? + ?? - 5 = 0.
Solution: Let the line ?? + ?? = 5 divides ???? in the ratio ?? : 1 at ??
? ???? - ???????????????? ???? ?? ?????? (
-3?? + 1
?? + 1
,
4?? + 2
?? + 1
) ?????????? ?? ???????? ???? ?? + ?? - 5 = 0 ?
-3?? + 1
?? + 1
+
4?? + 2
?? + 1
- 5
= 0 ? ?? = -
1
2
? Required ratio is 1: 2 externally. .
Aliter: Let the ratio is ?? : ??
?
?? ?? = -
(1 × 1 + 1 × 2 - 5)
1 × (-3) + 1 × 4 - 5
= -
1
2
? ?????????? ???? 1: 2 ???????????????????? .
Slope Formula:
If ?? is the angle at which a straight line is inclined to the positive direction of ?? -axis, & 0
°
= ?? <
180
°
, ?? ? 90
°
, then the slope of the line, denoted by ?? , is defined by ?? = ?????? ?? . If ?? is 90
°
, ?? does not
exist, but the line is parallel to the ?? -axis. If ?? = 0, then ?? = 0& the line is parallel to the ?? -axis. If
?? (?? 1
, ?? 1
)&?? (?? 2
, ?? 2
), ?? 1
? ?? 2
, are points on a straight line, then the slope ?? of the line is given by: ?? =
(
?? 1
-?? 2
?? 1
-?? 2
).
Problem 5: What is the slope of a line whose inclination with the positive direction of ?? -axis is:
(i) 30
°
(ii) 90
°
(iii)
135
°
Solution:
(i) Here ?? = 30
°
?????????? = ?????? ?? = ?????? 30
°
=
1
v3
(ii) Here ?? = 90
°
? The slope of line is not defined.
(iii) Here ?? = 135
°
? Slope = ?????? ?? = ?????? 135
°
= ?????? (180
°
- 45
°
) = -?????? 45
°
= -1.
Problem 6: Find the slope of the line passing through the points:
(i) (2,7) and (-3,4)
(ii)
(6,9) and (-2,7)
(i) Let ?? = (2,7) and ?? = (-3,4)
? Slope of ???? =
4-7
-3-2
=
3
5
Page 5
Straight Line
The knowledge of which geometry aims is the knowledge of the eternal...... Plato
A french mathematician and a greatest philosopher named Rene Descartes, pioneered the use of
algebra in Geometry. He suggested methods to study geometry by algebraic methods without making
direct reference to the actual figures
This geometry was called co-ordinate geometry or analytical geometry and it is the branch of geometry
in which algebraic equations are used to denote points, lines and curves.
Rectangular cartesian co-ordinate systems:
We shall right now focus on two-dimensional co-ordinate geometry in which two perpendicular lines
called co-ordinate axes (x-axis and ?? -axis) are used to locate a point in the plane.
?? is called origin. Any point ?? in this plane can be represented by a unique ordered pair ( ?? , ?? ), which
are called co-ordinates of that point. ?? is called ?? co-ordinate or abscissa and ?? is called ?? co-ordinate or
ordinate. The two perpendicular lines xox' and yoy' divide the plane in four regions which are called
quadrants, numbered as shown in the figure.
Let us look at some of the formulae linked with points now.
Distance Formula:
In rectangular Cartesian coordinate system
The distance between the points ?? (?? 1
, ?? 1
) and ?? (?? 2
, ?? 2
) is = v(?? 1
- ?? 2
)
2
+ (?? 1
- ?? 2
)
2
.
Problem1: Find the value of ?? , if the distance between the points (?? , 8) and (4,3) is 13
Solution: Let ?? (?? , 8) and ?? (4,3) be the given points. Then ???? = 13 (given)
v(?? - 4)
2
+ (8 - 3)
2
= 13 ? (?? - 4)
2
+ 25 = 169 ? ?? = 16 ???? ?? = -8
Section Formula:
If ?? (?? , ?? ) divides the line joining ?? (?? 1
, ?? 1
) & ?? (?? 2
, ?? 2
) in the ratio ?? : ?? , then;
?? =
?? ?? 2
+ ?? ?? 1
?? + ?? ; ?? =
?? ?? 2
+ ?? ?? 1
?? + ??
Notes: (i) If
?? ?? is positive, the division is internal, but if
?? ?? is negative, the division is external.
(ii) If ?? divides ???? internally in the ratio ?? : ?? &?? divides ???? externally in the ratio ?? : ?? then ?? &?? are
said to be harmonic conjugate of each other w.r.t. AB.
Mathematically,
2
????
=
1
????
+
1
????
i.e. ???? , ???? &???? are in H.P.
Problem 2: Find the co-ordinates of the point which divides the line segment joining the points (2,5)
and (-3,7) in the ratio 2: 3 (i) internally and (ii) externally.
Solution: Let ?? (?? , ?? ) be the required point.
(i) For internal division:
(?? , ?? ) = (
-6 + 6
2 + 3
,
14 + 15
2 + 3
) = (0,
29
5
)
(ii) For external division
(?? , ?? ) = (
-6 - 6
2 - 3
,
14 - 15
2 - 3
) = (12,1)
Problem 3: Find the co-ordinates of points which trisect the line segment joining (2, -3) and (4,5).
Solution: Let ?? (2, -3) and ?? (4,5) be the given points. Let the points of trisection be ?? and ?? . Then ???? =
???? = ???? = ?? (say)
? ???? = ???? + ???? = 2?? and ???? = ???? + ???? = 2??
? ???? : ???? = ?? : 2?? = 1: 2 and ???? : ???? = 2?? : ?? = 2: 1
So ?? divides ???? internally in the ratio 1: 2 while ?? divides internally in the ratio 2: 1
So ?? divides ???? internally in the ratio 1: 2 while ?? divides internally in the ratio 2: 1
? the co-ordinates of ?? are (
4+4
1+2
,
5-6
1+2
) = (
8
3
, -
1
3
)
and the co-ordinates of ?? are (
8+2
1+2
,
10-3
1+2
) = (
10
3
,
7
3
)
Hence, the points of trisection are (
8
3
, -
1
3
) and (
10
3
,
7
3
)
The ratio in which a given line divides the line segment
joining two points:
Let the given line ???? + ???? + ?? = 0 divide the line segment joining ?? (?? 1
, ?? 1
)&?? (?? 2
, ?? 2
) in the ratio ?? : ?? ,
then
?? ?? = -
?? ?? 1
+?? ?? 1
+?? ?? ?? 2
+?? ?? 2
+?? . If ?? &?? are on the same side of the given line then ?? /?? is negative but if ?? & ?? are
on opposite sides of the given line, then ?? /?? is positive
Example# 4: Find the ratio in which the line joining the points ?? (1,2) and ?? (-3,4) is divided by the line
?? + ?? - 5 = 0.
Solution: Let the line ?? + ?? = 5 divides ???? in the ratio ?? : 1 at ??
? ???? - ???????????????? ???? ?? ?????? (
-3?? + 1
?? + 1
,
4?? + 2
?? + 1
) ?????????? ?? ???????? ???? ?? + ?? - 5 = 0 ?
-3?? + 1
?? + 1
+
4?? + 2
?? + 1
- 5
= 0 ? ?? = -
1
2
? Required ratio is 1: 2 externally. .
Aliter: Let the ratio is ?? : ??
?
?? ?? = -
(1 × 1 + 1 × 2 - 5)
1 × (-3) + 1 × 4 - 5
= -
1
2
? ?????????? ???? 1: 2 ???????????????????? .
Slope Formula:
If ?? is the angle at which a straight line is inclined to the positive direction of ?? -axis, & 0
°
= ?? <
180
°
, ?? ? 90
°
, then the slope of the line, denoted by ?? , is defined by ?? = ?????? ?? . If ?? is 90
°
, ?? does not
exist, but the line is parallel to the ?? -axis. If ?? = 0, then ?? = 0& the line is parallel to the ?? -axis. If
?? (?? 1
, ?? 1
)&?? (?? 2
, ?? 2
), ?? 1
? ?? 2
, are points on a straight line, then the slope ?? of the line is given by: ?? =
(
?? 1
-?? 2
?? 1
-?? 2
).
Problem 5: What is the slope of a line whose inclination with the positive direction of ?? -axis is:
(i) 30
°
(ii) 90
°
(iii)
135
°
Solution:
(i) Here ?? = 30
°
?????????? = ?????? ?? = ?????? 30
°
=
1
v3
(ii) Here ?? = 90
°
? The slope of line is not defined.
(iii) Here ?? = 135
°
? Slope = ?????? ?? = ?????? 135
°
= ?????? (180
°
- 45
°
) = -?????? 45
°
= -1.
Problem 6: Find the slope of the line passing through the points:
(i) (2,7) and (-3,4)
(ii)
(6,9) and (-2,7)
(i) Let ?? = (2,7) and ?? = (-3,4)
? Slope of ???? =
4-7
-3-2
=
3
5
( ?????????? ?????????? =
?? 2
- ?? 1
?? 2
- ?? 1
)
(ii) Let ?? = (6,9), ?? = (-2,7)
? Slope of ???? =
7-9
-2-6
=
1
4
Condition of collinearity of three points:
Points ?? (?? 1
, ?? 1
), ?? (?? 2
, ?? 2
), ?? (?? 3
, ?? 3
) are collinear if
?? ????
= ?? ????
= ?? ????
?? . ?? . (
?? 1
- ?? 2
?? 1
- ?? 2
) = (
?? 2
- ?? 3
?? 2
- ?? 3
) #(?? )
(iii) ???? = ???? + ???? or ???? ~ ????
(iv) A divides the line segment ???? in some ratio.
Example# 7: Show that the points (-2, -1), (2,7) and (5,13) are collinear.
Solution: Let (-2, -1)(2,7) and (5,13) be the co-ordinates of the points ?? , ?? and ?? respectively.
Slope of ???? =
7+1
2+2
= 2 and Slope of ???? =
13-7
5-2
= 2
? Slope of ???? = slope of ????
? ???? &???? are parallel
? ?? , ?? , ?? are collinear because ?? is on both lines ???? and ???? .
Area of a Triangle:
If ?? (?? 1
, ?? 1
), ?? (?? 2
, ?? 2
), ?? (?? 3
, ?? 3
) are the vertices of triangle ?????? , then its area is equal to
?? ?????? =
1
2
|?? 1
?? 1
1 ?? 2
?? 2
1 ?? 3
?? 3
1 |, provided the vertices are considered in the counter clockwise sense.
The
above formula will give ?? (-) ve area if the vertices (?? ?? , ?? ?? ), ?? = 1,2,3 are placed in the clockwise sense.
Note: Area of ?? -sided polygon formed by points (?? 1
, ?? 1
); (?? 2
, ?? 2
); … … . ; (?? ?? , ?? ?? ) is given by
1
2
(|?? 1
?? 2
?? 1
?? 2
| + |?? 2
?? 3
?? 2
?? 3
| + ? … … … . |?? ?? -1
?? ?? ?? ?? -1
?? ?? | + |?? ?? ?? 1
?? ?? ?? 1
|)
Here vertices are taken in order.
Problem 8: If the co-ordinates of two points ?? and ?? are (2,1) and (4, -3) respectively. Find the
coordinates of any point ?? if ???? = ???? and ?? rea of ? ?????? = 6.
Solution: Let the co-ordinates of ?? be (?? , ?? ). Then
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