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Straight Lines Class 11 Notes Maths Chapter 9

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 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 
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FAQs on Straight Lines Class 11 Notes Maths Chapter 9

1. What are the properties of a straight line in geometry?
Ans. In geometry, a straight line is defined by the following properties: - It has infinite length and no thickness. - It extends in both directions indefinitely. - It is the shortest distance between two points. - It can be represented by an equation of the form y = mx + c, where m is the slope and c is the y-intercept.
2. How do you find the slope of a straight line given its equation?
Ans. To find the slope of a straight line given its equation in the form y = mx + c, the coefficient of x (m) represents the slope. The slope indicates the steepness or the incline of the line. If the line is upward sloping, m will be positive, and if the line is downward sloping, m will be negative.
3. What is the significance of the y-intercept in the equation of a straight line?
Ans. The y-intercept (c) in the equation of a straight line represents the point where the line intersects the y-axis. It is the value of y when x is equal to 0. The y-intercept helps determine the starting point of the line on the y-axis and gives information about the position of the line in the coordinate plane.
4. How can the distance between a point and a straight line be calculated?
Ans. The distance between a point (x₁, y₁) and a straight line ax + by + c = 0 can be calculated using the formula: Distance = |ax₁ + by₁ + c| / √(a² + b²) This formula is derived from the perpendicular distance formula, where the distance is the length of the perpendicular segment drawn from the point to the line.
5. How do you determine whether two straight lines are parallel or perpendicular?
Ans. Two straight lines are parallel if their slopes are equal. In other words, if the equations of the lines are in the form y = mx + c, then the slopes (m) of the lines should be the same. Two straight lines are perpendicular if the product of their slopes is -1. In other words, if the equations of the lines are in the form y = mx + c, then the slopes (m₁ and m₂) of the lines should satisfy the equation m₁ * m₂ = -1.
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