Class 11 Maths Revision Notes for Chapter-10 Straight Lines


Class 11 Maths Revision Notes for Chapter-10 Straight Lines

Class 11 Maths Revision Notes for Chapter-10 Straight Lines - Free PDF Download

Free PDF download of Class 11 Maths revision notes & short key-notes for Chapter-10 Straight Lines to score high marks in exams, prepared by expert mathematics teachers from latest edition of CBSE books.

CBSE Class 11 Mathematics Revision Notes Chapter-10
Straight Lines


  1. Slope of a Line
  2. Various Forms of the Equation of a Line
  3. General Equation of a Line and Distance of a Point From a Line

First Degree Equation

Every first degree equation like ax+by+c=0 would be the equation of a straight line.

Slope of a line

  • Slope (m) of a non-vertical line passing through the points (x1 , y1 ) and (x2 , y2) is given by is given by =y1y2x1x2 x1x2.
  • If a line makes an angle á with the positive direction of x-axis, then the slope of the line is given by m =tanαα90o 
  • Slope of horizontal line is zero and slope of vertical line is undefined.
  • An acute angle (say θ) between lines L1 and L2 with slopes m1 and m2 is given by tanθ=|m2m11+m1m2|, 1+m1m20
  • Two lines are parallel if and only if their slopes are equal i.e., m1=m2
  • Two lines are perpendicular if and only if product of their slopes is –1, i.e., m1.m2=1
  • Three points A, B and C are collinear, if and only if slope of AB = slope of BC.
  • Equation of the horizontal line having distance a from the x-axis is eithery = a or y = – a.
  • Equation of the vertical line having distance b from the y-axis is eitherx = b or x = – b.
  • The point (x, y) lies on the line with slope m and through the fixed point (xo, y0 ), if and only if its coordinates satisfy the equation.

Various forms of equations of a line:

  • Two points form: Equation of the line passing through the points (x1, y1) and ((x2, y2) is given by yy1=y2y1x2x1(xx1)
  • Slope-Intercept form: The point (x, y) on the line with slope m and y-intercept c lies on the line if and only if y=mx +c.
  • If a line with slope m makes x-intercept d. Then equation of the line is y=m(x -d).
  • Intercept form: Equation of a line making intercepts a and b on the x-and y-axis, respectively, is xa+yb=1.
  • Normal form: The equation of the line having normal distance from origin p and angle between normal and the positive xaxis ω is given by  x cosω +ysin ω=p
  • General Equation of a Line: Any equation of the form Ax + By + C = 0, with A and B are not zero, simultaneously, is called the general linear equation or general equation of a line.
  • Working Rule for reducing general form into the normal form:

(i) Shift constant "C" to the R.H.S. and get Ax+By=C
(ii) If the R.H.S. is not positive, then make it positive by multiplying the whole equation by -1.
(iii) Divide both sides of equation by A2+B2.

The equation so obtained is in the normal form.

  • Parametric Equation (Symmetric Form): xx1cosθ=yy1sinθ=r
  • Equation of a line through origin: y=mx or y=xtanθ.
  • The perpendicular distance (d) of a line Ax + By+ C = 0 from a point   (x1, y1) is given by d=|Ax1+By1+C|A2+B2
  • Distance between the parallel lines Ax + By + C1= 0 and Ax + By + C2= 0, is given by d=|C1C2|A2+B2

Concurrent Lines
Three of more straight lines are said to be concurrent if they pass through a common point i.e., they meet at a point. Thus, if three lines are concurrent the point of intersection of two lines lies on the third line.

Condition of concurrency of three lines:
a1(b2c3b3c2)+b1(c2a3c3a2)+c1(a2b3a3b2)=0

EQUATIONS OF FAMILY OF LINES THROUGH THE INTERSECTION OF TWO LINES
A1x+B1y+C1+k(A2x+B2y+C2)=0
where k is a constant and also called parameter.
This equation is of first degree of x and y, therefore, it represents a family of lines.

DISTANCE BETWEEN TWO PARALLEL LINES
Working Rule to find the distance between two parallel lines:
(i) Find the co-ordinates of any point on one of ht egiven line, preferably by putting x=0 and y=0.
(ii) The perpendicular distance of this point from the other line is the required distance between the lines.

CBSE Class 11 Mathematics Revision Notes Chapter-10
STRAIGHT LINE


Definition: A straight line is a curve such that every point on the line segment joining any two points on It lies on it. (No turning point b/w two points called a straight line)

Slope of Line (Gradient): A line makes with the +ve direction of the x – axis in anticlockwise sense is Called the slope or gradient of the line.

The slope of a line is generally denoted by m. Thus, m = tanθ.

  1. Since a line parallel to x –axis makes an angle of 00 with x – axis, therefore its slope is tan 0° = 0.
  2. A line parallel to y – axis makes an angle of 90° with x – axis, so its slope is tanπ2=.

Slope of Line when Passing from two given points:
If P(x1, y1) & (x2, y2) So, m=y2y1x2x1

Angle between two Lines:

tanθ=|m2m11+m1m2| here m1:m2 are slope of lines and θ is angle bw two lines.

NOTE: 1. If two lines are parallel to each other ⇒ m1 = m2 because θ=0

  1. if two line are perpendicular to each other ⇒ m1m2 = -1 because θ=90
  2. if line parallel to x - axis ⇒ equation of line y = k
  3. if line parallel to y - axis ⇒ equation of line x = k
  4. every linear equation of two variable represent a line e.g. ax + by c = 0

Intercepts of line on the Axes:

B Here OA = X axis intercepts
And OB = Y axis intercepts
Let OA = a and OB = b
So, A(a, 0) and B(0, b)
NOTE: If three point are collinear than slope are equal b/w any two point of line
let A(x1,y1): B(x2,y2) & (x3,y3) ⇒ slope of BC = slope of AC

Different forms of the equation of a straight line:

  1. Slope intercept form of a line:
    The equation of a line with slope m and making an intercept c on y – axis is y = mx + c

    The equation of a line with slope m and making an intercept c on x – axis is y = m(x - c)
  2. Point - slope form of a line:
    The equation of a line which passes through the point (given) P(x1, y1) and has the slope ‘m’ is
    y - y1 = m(x - x1).
  3. Two point form of a line:
    The equation of a line passing through two points P(x1, y2) and Q(x2, y2) is
    yy1=y2y1x2x1(xx1)
  4. Intercept form of a line:
    The equation of a line which cuts off intercepts ‘a’ and ‘b’ respectively from the x – axis and y – axis is xa+yb=1.
  5. Normal form or Perpendicular form of a line:
    The equation of the straight line upon which the length of the perpendicular from the origin is p and this Perpendicular makes an angle α with x – axis is xcosα+ysinα=p
  6. Distance form of a line:
    The equation of the straight line passing through (x1, y1) and making an angle θ with the +ve direction of x – axis is xx1cosθ=yy1sinθ=r
    Where r is the distance of the point (x, y) on the line from the point (x1, y1)

Transformation of general equation in different standard forms:

  1. Transformation of Ax + By + C = 0 in the slope intercept form y = m x + c
    y=(AB)x+(CB)
    This is of the form y = m x +c, where
    m=AB=cofficientofxcofficientofy, and intercept on y – axis =CB=constentcofficientofy
  2. Transformation of Ax + By + C = 0 in intercept form xa+yb=1
    x(CA)+y(CB)=1
    Intercept on x – axis =CA=constanttermcofficientofx, Intercept on y - axis =CB=constanttermcofficientofy
  3. Transformation of Ax + By + C = 0 in intercept form xcosα+ysinα=p
    AA2+B2xBA2+B2y=CA2+B2
    Here cosα=AA2+B2 and sinα=BA2+B2;p=±CA2+B2
    NOTE: Three lines are said to be concurrent if they pass through a common point OR they meet at a point.

Lines parallel and Perpendicular to a given line:

  1. Line parallel to a guven line
    To write a line parallel to a given line we keep the expression containing x and y same and simply replace The given constant by an unknown constant k. the value of k can be determined by some given condition.
  2. Line perpendicular to a guven line
    The equation of a line perpendicular to a given line ax + by + c = 0 is bx – ay + k = 0.

Distance of a point from a line:
The length of the perpendicular from a point (α,β) to a line ax + by + c = 0 is
Distance B/W Parallel lines:
The distance between two parallel lines ax + by +c1 = 0 and ax + by + c2 = 0 is
|c1c2a2+b2|