# 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 and is given by is given by $x1≠x2$.
• If a line makes an angle á with the positive direction of x-axis, then the slope of the line is given by
• Slope of horizontal line is zero and slope of vertical line is undefined.
• An acute angle (say θ) between lines with slopes is given by $tan⁡θ=|m2−m11+m1m2|$, $1+m1m2≠0$
• 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 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 and ( is given by $y−y1=y2−y1x2−x1(x−x1)$
• 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 is given by
• 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): $x−x1cos⁡θ=y−y1sin⁡θ=r$
• Equation of a line through origin: $y=mx$ or $y=xtan⁡θ$.
• The perpendicular distance (d) of a line Ax + By+ C = 0 from a point is given by $d=|Ax1+By1+C|A2+B2$
• Distance between the parallel lines = 0 and = 0, is given by $d=|C1−C2|A2+B2$

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:
$a1(b2c3−b3c2)+b1(c2a3−c3a2)+c1(a2b3−a3b2)=0$

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 = $tan⁡θ$.

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 $tan⁡π2=∞$.

Slope of Line when Passing from two given points:
If P(x1, y1) & (x2, y2) So, $m=y2−y1x2−x1$

Angle between two Lines:

$tan⁡θ=|m2−m11+m1m2|$ 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
$y−y1=y2−y1x2−x1(x−x1)$
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 $xcos⁡α+ysin⁡α=p$
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 $x−x1cos⁡θ=y−y1sin⁡θ=r$
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 $=−CA=−constan⁡ttermcofficientofx$, Intercept on y - axis $=−CB=−constan⁡ttermcofficientofy$
3. Transformation of Ax + By + C = 0 in intercept form $xcos⁡α+ysin⁡α=p$
$−AA2+B2x−BA2+B2y$$=CA2+B2$
Here $cos⁡α=−AA2+B2$ and $sin⁡α=−BA2+B2;p=±CA2+B2$
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
$|c1−c2a2+b2|$