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=0ax+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 )(x1,y1) and (x2 , y2)(x2,y2) is given by is given by m =y1−y2x1−x2= m =y1−y2x1−x2= x1≠x2x1≠x2.

• If a line makes an angle á with the positive direction of x-axis, then the slope of the line is given by m =tanα, α≠90o m =tanα, α≠90o

• Slope of horizontal line is zero and slope of vertical line is undefined.
• An acute angle (say θ) between lines L1 and L2L1 and L2 with slopes m1 and m2m1 and m2 is given by tanθ=∣∣m2−m11+m1m2∣∣tan⁡θ=|m2−m11+m1m2|, 1+m1m2≠01+m1m2≠0

• Two lines are parallel if and only if their slopes are equal i.e., m1=m2m1=m2
• Two lines are perpendicular if and only if product of their slopes is –1, i.e., m1.m2=−1m1.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 ),(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)(x1,y1) and ((x2, y2)(x2,y2) is given by y−y1=y2−y1x2−x1(x−x1)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 +cy=mx +c.
• If a line with slope m makes x-intercept d. Then equation of the line is y=m(x -d)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=1xa+yb=1.
• Normal form: The equation of the line having normal distance from origin p and angle between normal and the positive x−axis ωx−axis ω is given by  x cosω +ysin ω=px 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=−CAx+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−−−−−−−√A2+B2.

The equation so obtained is in the normal form.

• Parametric Equation (Symmetric Form): x−x1cosθ=y−y1sinθ=rx−x1cos⁡θ=y−y1sin⁡θ=r
• Equation of a line through origin: y=mxy=mx or y=xtanθy=xtan⁡θ.
• The perpendicular distance (d) of a line Ax + By+ C = 0 from a point   (x1, y1)(x1,y1) is given by d=|Ax1+By1+C|A2+B2√d=|Ax1+By1+C|A2+B2
• Distance between the parallel lines Ax + By + C1Ax + By +C1= 0 and Ax + By + C2Ax + By +C2= 0, is given by d=|C1−C2|A2+B2√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)=0a1(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)=0A1x+B1y+C1+k(A2x+B2y+C2)=0
where kk is a constant and also called parameter.
This equation is of first degree of xx and yy, 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=0x=0 and y=0y=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 = tanθtan⁡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 tanπ2=∞tan⁡π2=∞.

Slope of Line when Passing from two given points:
If P(x₁, y₁) & (x₂, y₂) So, m=y2−y1x2−x1m=y2−y1x2−x1

Angle between two Lines:

tanθ=∣∣m2−m11+m1m2∣∣tan⁡θ=|m2−m11+m1m2| here m1:m2 are slope of lines and θθ is angle bwbw two lines.

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

1. if two line are perpendicular to each other ⇒ m₁m₂ = -1 because θ=90θ=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)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=1xa+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 αα with x – axis is xcosα+ysinα=pxcos⁡α+ysin⁡α=p
   
6. Distance form of a line:
   The equation of the straight line passing through (x₁, y₁) and making an angle θθ with the +ve direction of x – axis is x−x1cosθ=y−y1sinθ=rx−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)y=(−AB)x+(−CB)
   This is of the form y = m x +c, where
   m=−AB=−cofficientofxcofficientofym=−AB=−cofficientofxcofficientofy, and intercept on y – axis =−CB=−constentcofficientofy=−CB=−constentcofficientofy
2. Transformation of Ax + By + C = 0 in intercept form xa+yb=1xa+yb=1
   x(−CA)+y(−CB)=1x(−CA)+y(−CB)=1
   Intercept on x – axis =−CA=−constanttermcofficientofx=−CA=−constan⁡ttermcofficientofx, Intercept on y - axis =−CB=−constanttermcofficientofy=−CB=−constan⁡ttermcofficientofy
3. Transformation of Ax + By + C = 0 in intercept form xcosα+ysinα=pxcos⁡α+ysin⁡α=p
   −AA2+B2√x−BA2+B2√y−AA2+B2x−BA2+B2y=CA2+B2√=CA2+B2
   Here cosα=−AA2+B2√cos⁡α=−AA2+B2 and sinα=−BA2+B2√;p=±CA2+B2√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)(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√∣∣∣|c1−c2a2+b2|