Straight Lines Class 11 Notes Maths Chapter 9

 Page 1


 
KEY POINTS
? Distance between two points A(x1, y1) and B (x2, y2) is given by 
? ? ? ?
2 2
2 1 2 1
– AB x x y y ? ? ?
? Let the vertices of a triangle ABC are A(x1, y1) B (x2, y2) and 
C(x3, y3).Then area of triangle  
? ? ? ? ? ?
1 2 3 2 3 1 3 1 2
1
–
2
ABC x y y x y y x y y ? ? ? ? ?
? NOTE: Area of a ? is always positive. If the above expression 
is zero, then a ? is not possible. Thus the points are collinear. 
? LOCUS: When a variable point P(x,y) moves under certain 
condition then the path traced out by the point P is called the 
locus of the point. 
For example: Locus of a point P, which moves such that its 
distance from a fixed point C is always constant, is a circle. 
Page 2


 
KEY POINTS
? Distance between two points A(x1, y1) and B (x2, y2) is given by 
? ? ? ?
2 2
2 1 2 1
– AB x x y y ? ? ?
? Let the vertices of a triangle ABC are A(x1, y1) B (x2, y2) and 
C(x3, y3).Then area of triangle  
? ? ? ? ? ?
1 2 3 2 3 1 3 1 2
1
–
2
ABC x y y x y y x y y ? ? ? ? ?
? NOTE: Area of a ? is always positive. If the above expression 
is zero, then a ? is not possible. Thus the points are collinear. 
? LOCUS: When a variable point P(x,y) moves under certain 
condition then the path traced out by the point P is called the 
locus of the point. 
For example: Locus of a point P, which moves such that its 
distance from a fixed point C is always constant, is a circle. 
 
CP = constant 
? Locus of an equation: In the coordinate plane, locus of an 
equation is the pictorial representation of the set of all those 
points which satisfy the given equation. 
? Equation of a locus: is the equation in x and y that is satisfied by 
the coordinates of every point on the locus. 
? A line is also defined as the locus of a point satisfying the 
condition ax + by + c = 0 where a, b, c are constants. 
? Slope of a straight line: If ? is the inclination of a line thentan ? is 
defined as slope of the straight line L and denoted by m 
tan , 90 m ? ? ? ? ?
If 0 90 0 then m and ? ? ? ? ? ?
90 180 0 then m ? ? ? ? ? ? 
? NOTE 1: The slope of a line whose inclination is 90° is not 
defined. Slope of x-axis is zero and slope of y-axis is not defined 
? NOTE 2: Slope of any horizontal line i.e. || to x-axis is 
zero.Slope of a vertical line i.e. || to y-axis is not zero. 
Page 3


 
KEY POINTS
? Distance between two points A(x1, y1) and B (x2, y2) is given by 
? ? ? ?
2 2
2 1 2 1
– AB x x y y ? ? ?
? Let the vertices of a triangle ABC are A(x1, y1) B (x2, y2) and 
C(x3, y3).Then area of triangle  
? ? ? ? ? ?
1 2 3 2 3 1 3 1 2
1
–
2
ABC x y y x y y x y y ? ? ? ? ?
? NOTE: Area of a ? is always positive. If the above expression 
is zero, then a ? is not possible. Thus the points are collinear. 
? LOCUS: When a variable point P(x,y) moves under certain 
condition then the path traced out by the point P is called the 
locus of the point. 
For example: Locus of a point P, which moves such that its 
distance from a fixed point C is always constant, is a circle. 
 
CP = constant 
? Locus of an equation: In the coordinate plane, locus of an 
equation is the pictorial representation of the set of all those 
points which satisfy the given equation. 
? Equation of a locus: is the equation in x and y that is satisfied by 
the coordinates of every point on the locus. 
? A line is also defined as the locus of a point satisfying the 
condition ax + by + c = 0 where a, b, c are constants. 
? Slope of a straight line: If ? is the inclination of a line thentan ? is 
defined as slope of the straight line L and denoted by m 
tan , 90 m ? ? ? ? ?
If 0 90 0 then m and ? ? ? ? ? ?
90 180 0 then m ? ? ? ? ? ? 
? NOTE 1: The slope of a line whose inclination is 90° is not 
defined. Slope of x-axis is zero and slope of y-axis is not defined 
? NOTE 2: Slope of any horizontal line i.e. || to x-axis is 
zero.Slope of a vertical line i.e. || to y-axis is not zero. 
 
? Three points A, B and C lying in a plane are collinear, if slope of 
AB = Slope of BC. 
? Slope of a line through given points (x1, y1) and (x2,y2) is given 
by 
2 1
2 1
y y
m
x x
?
?
?
. 
? Two lines are parallel to each other if and only if their slopes are 
equal. 
1 2 1 2
. || i e l l m m ? ?
? NOTE: If slopes of lines 
1
l and 
2
l are not defined then they must 
be ? to x-axis, so they are ||. Thus 
1 2
|| l l ? they have same 
slope or both of them have no slope. 
? Two non- vertical lines are perpendicular to each other if and 
only if their slopes are negative reciprocal of each other. 
1 2 1 2
. 1 ie l l m m ? ? ? ? 
? NOTE: The above condition holds when the lines have non-zero 
slopes i.e none  of them ? to any axis. 
? Acute angle ? between two lines, whose slopes are m1 and m2 
is given by 
?
1 2
1 2
m m
tan ,
1 m m
?
?
?
1 + m
1
m
2
 ? 0
& obtuse angle is 180 ? ? ? ? 
Page 4


 
KEY POINTS
? Distance between two points A(x1, y1) and B (x2, y2) is given by 
? ? ? ?
2 2
2 1 2 1
– AB x x y y ? ? ?
? Let the vertices of a triangle ABC are A(x1, y1) B (x2, y2) and 
C(x3, y3).Then area of triangle  
? ? ? ? ? ?
1 2 3 2 3 1 3 1 2
1
–
2
ABC x y y x y y x y y ? ? ? ? ?
? NOTE: Area of a ? is always positive. If the above expression 
is zero, then a ? is not possible. Thus the points are collinear. 
? LOCUS: When a variable point P(x,y) moves under certain 
condition then the path traced out by the point P is called the 
locus of the point. 
For example: Locus of a point P, which moves such that its 
distance from a fixed point C is always constant, is a circle. 
 
CP = constant 
? Locus of an equation: In the coordinate plane, locus of an 
equation is the pictorial representation of the set of all those 
points which satisfy the given equation. 
? Equation of a locus: is the equation in x and y that is satisfied by 
the coordinates of every point on the locus. 
? A line is also defined as the locus of a point satisfying the 
condition ax + by + c = 0 where a, b, c are constants. 
? Slope of a straight line: If ? is the inclination of a line thentan ? is 
defined as slope of the straight line L and denoted by m 
tan , 90 m ? ? ? ? ?
If 0 90 0 then m and ? ? ? ? ? ?
90 180 0 then m ? ? ? ? ? ? 
? NOTE 1: The slope of a line whose inclination is 90° is not 
defined. Slope of x-axis is zero and slope of y-axis is not defined 
? NOTE 2: Slope of any horizontal line i.e. || to x-axis is 
zero.Slope of a vertical line i.e. || to y-axis is not zero. 
 
? Three points A, B and C lying in a plane are collinear, if slope of 
AB = Slope of BC. 
? Slope of a line through given points (x1, y1) and (x2,y2) is given 
by 
2 1
2 1
y y
m
x x
?
?
?
. 
? Two lines are parallel to each other if and only if their slopes are 
equal. 
1 2 1 2
. || i e l l m m ? ?
? NOTE: If slopes of lines 
1
l and 
2
l are not defined then they must 
be ? to x-axis, so they are ||. Thus 
1 2
|| l l ? they have same 
slope or both of them have no slope. 
? Two non- vertical lines are perpendicular to each other if and 
only if their slopes are negative reciprocal of each other. 
1 2 1 2
. 1 ie l l m m ? ? ? ? 
? NOTE: The above condition holds when the lines have non-zero 
slopes i.e none  of them ? to any axis. 
? Acute angle ? between two lines, whose slopes are m1 and m2 
is given by 
?
1 2
1 2
m m
tan ,
1 m m
?
?
?
1 + m
1
m
2
 ? 0
& obtuse angle is 180 ? ? ? ? 
 
? x = a is a line parallel to y-axis at a distance of a units from y-
axis. x = a lies on right or left of y-axis according as a is positive 
or negative. 
? y = b is a line parallel to x-axis at a distance of ‘b’ units fromx-
axis. y=b lies above or below x-axis, according as b is positive or 
negative. 
? Point slope form 
? Equation of a line passing through given point (x1, y1) and 
having slope m is given by y – y1 = m(x – x1) 
Page 5


 
KEY POINTS
? Distance between two points A(x1, y1) and B (x2, y2) is given by 
? ? ? ?
2 2
2 1 2 1
– AB x x y y ? ? ?
? Let the vertices of a triangle ABC are A(x1, y1) B (x2, y2) and 
C(x3, y3).Then area of triangle  
? ? ? ? ? ?
1 2 3 2 3 1 3 1 2
1
–
2
ABC x y y x y y x y y ? ? ? ? ?
? NOTE: Area of a ? is always positive. If the above expression 
is zero, then a ? is not possible. Thus the points are collinear. 
? LOCUS: When a variable point P(x,y) moves under certain 
condition then the path traced out by the point P is called the 
locus of the point. 
For example: Locus of a point P, which moves such that its 
distance from a fixed point C is always constant, is a circle. 
 
CP = constant 
? Locus of an equation: In the coordinate plane, locus of an 
equation is the pictorial representation of the set of all those 
points which satisfy the given equation. 
? Equation of a locus: is the equation in x and y that is satisfied by 
the coordinates of every point on the locus. 
? A line is also defined as the locus of a point satisfying the 
condition ax + by + c = 0 where a, b, c are constants. 
? Slope of a straight line: If ? is the inclination of a line thentan ? is 
defined as slope of the straight line L and denoted by m 
tan , 90 m ? ? ? ? ?
If 0 90 0 then m and ? ? ? ? ? ?
90 180 0 then m ? ? ? ? ? ? 
? NOTE 1: The slope of a line whose inclination is 90° is not 
defined. Slope of x-axis is zero and slope of y-axis is not defined 
? NOTE 2: Slope of any horizontal line i.e. || to x-axis is 
zero.Slope of a vertical line i.e. || to y-axis is not zero. 
 
? Three points A, B and C lying in a plane are collinear, if slope of 
AB = Slope of BC. 
? Slope of a line through given points (x1, y1) and (x2,y2) is given 
by 
2 1
2 1
y y
m
x x
?
?
?
. 
? Two lines are parallel to each other if and only if their slopes are 
equal. 
1 2 1 2
. || i e l l m m ? ?
? NOTE: If slopes of lines 
1
l and 
2
l are not defined then they must 
be ? to x-axis, so they are ||. Thus 
1 2
|| l l ? they have same 
slope or both of them have no slope. 
? Two non- vertical lines are perpendicular to each other if and 
only if their slopes are negative reciprocal of each other. 
1 2 1 2
. 1 ie l l m m ? ? ? ? 
? NOTE: The above condition holds when the lines have non-zero 
slopes i.e none  of them ? to any axis. 
? Acute angle ? between two lines, whose slopes are m1 and m2 
is given by 
?
1 2
1 2
m m
tan ,
1 m m
?
?
?
1 + m
1
m
2
 ? 0
& obtuse angle is 180 ? ? ? ? 
 
? x = a is a line parallel to y-axis at a distance of a units from y-
axis. x = a lies on right or left of y-axis according as a is positive 
or negative. 
? y = b is a line parallel to x-axis at a distance of ‘b’ units fromx-
axis. y=b lies above or below x-axis, according as b is positive or 
negative. 
? Point slope form 
? Equation of a line passing through given point (x1, y1) and 
having slope m is given by y – y1 = m(x – x1) 
 
? Two point form 
? Equation of a line passing through given points (x1 , y1) and 
(x2, y2) is given by ? ?
2 1
1 1
2 1
y y
y y x x
x x
?
? ? ?
?
? Slope intercept form(y - intercept) 
? Equation of a line having slope m and y-intercept ‘c’ is given by 
y = mx + c