Straight Lines - Revision Notes

 CBSE Class 11 Mathematics

Revision Notes

Chapter-10

Straight Lines

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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$ax+by+c=0$ 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 )$\left({\text{x}}_1,{\text{y}}_1\right)$ and (x2 , y2)$\left({\text{x}}_2,{\text{y}}_2\right)$ is given by is given by m =y1−y2x1−x2= $\text{m }=\frac{y_1-y_2}{x_1-x_2}\text{= }$ x1≠x2$x_1\ne x_2$.

• 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 $\text{m =tan}\alpha \text{, }\alpha \ne \text{9}{\text{0}}^o$

• Slope of horizontal line is zero and slope of vertical line is undefined.
• An acute angle (say θ) between lines L1 and L2${\text{L}}_1\text{ and }{\text{L}}_2$ with slopes m1 and m2${\text{m}}_1\text{ and }{\text{m}}_2$ is given by tanθ=∣∣m2−m11+m1m2∣∣$\tan \theta =\left|\frac{m_2-m_1}{1+m_1{m}_2}\right|$, 1+m1m2≠0$1+m_1{m}_2\ne 0$

• Two lines are parallel if and only if their slopes are equal i.e., m1=m2$m_1=m_2$
• Two lines are perpendicular if and only if product of their slopes is –1, i.e., m1.m2=−1$m_1.m_2=-1$
• 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 ),$\left({\text{x}}_o,{\text{y}}_0\right),$ 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)$\left({\text{x}}_1,{\text{y}}_1\right)$ and ((x2, y2)$\text{(}{\text{x}}_2,{\text{y}}_2)$ is given by y−y1=y2−y1x2−x1(x−x1)$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)$
• 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$y=\text{mx +c}$.
• If a line with slope m makes x-intercept d. Then equation of the line is y=m(x -d)$y=\text{m(x -d)}$.
• Intercept form: Equation of a line making intercepts a and b on the x-and y-axis, respectively, is xa+yb=1$\frac{x}{a}+\frac{y}{b}=1$.
• Normal form: The equation of the line having normal distance from origin p and angle between normal and the positive x−axis ω$\text{x}-\text{axis }\omega$ is given by  x cosω +ysin ω=p$\text{ x cos}\omega \text{ +ysin }\omega =p$
• 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$Ax+By=-C$
(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−−−−−−−√$\sqrt{{\mathrm{A}}^2+{\mathrm{B}}^2}$.

The equation so obtained is in the normal form.

• Parametric Equation (Symmetric Form): x−x1cosθ=y−y1sinθ=r$\frac{x-x_1}{\cos \theta }=\frac{y-y_1}{\sin \theta }=r$
• Equation of a line through origin: y=mx$y=mx$ or y=xtanθ$y=x\tan \theta$.
• The perpendicular distance (d) of a line Ax + By+ C = 0 from a point   (x1, y1)$\left({\text{x}}_1,{\text{y}}_1\right)$ is given by d=|Ax1+By1+C|A2+B2√$d=\frac{\left|A{x}_1+B{y}_1+C\right|}{\sqrt{A^2+B^2}}$
• Distance between the parallel lines Ax + By + C1$\text{Ax }+\text{ By }+{\text{C}}_1$= 0 and Ax + By + C2$\text{Ax }+\text{ By }+{\text{C}}_2$= 0, is given by d=|C1−C2|A2+B2√$d=\frac{\left|C_1-C_2\right|}{\sqrt{A^2+B^2}}$

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$a_1\left(b_2{c}_3-b_3{c}_2\right)+b_1\left(c_2{a}_3-c_3{a}_2\right)+c_1\left(a_2{b}_3-a_3{b}_2\right)=0$

EQUATIONS OF FAMILY OF LINES THROUGH THE INTERSECTION OF TWO LINES
A1x+B1y+C1+k(A2x+B2y+C2)=0${\mathrm{A}}_{1}x+{\mathrm{B}}_{1}y+{\mathrm{C}}_1+k\left({\mathrm{A}}_{2}x+{\mathrm{B}}_{2}y+{\mathrm{C}}_2\right)=0$
where k$k$ is a constant and also called parameter.
This equation is of first degree of x$x$ and y$y$, 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$x=0$ and y=0$y=0$.
(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

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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$\tan \theta$.

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=∞$\tan \frac{\pi }{2}=\mathrm{\infty }$.

Slope of Line when Passing from two given points:
If P(x1, y1) & (x2, y2) So, m=y2−y1x2−x1$m=\frac{y_2-y_1}{x_2-x_1}$

Angle between two Lines:

tanθ=∣∣m2−m11+m1m2∣∣$\tan \theta =\left|\frac{m_2-m_1}{1+m_1{m}_2}\right|$ here m1:m2 are slope of lines and θ$\theta$ is angle bw$\frac{b}{w}$ two lines.

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

1. if two line are perpendicular to each other ⇒ m1m2 = -1 because θ=90$\theta =90$
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)$y-y_1=\frac{y_2-y_1}{x_2-x_1}\left(x-x_1\right)$
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$\frac{x}{a}+\frac{y}{b}=1$.
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$\alpha$ with x – axis is xcosα+ysinα=p$x\cos \alpha +y\sin \alpha =p$
   
6. Distance form of a line:
   The equation of the straight line passing through (x1, y1) and making an angle θ$\theta$ with the +ve direction of x – axis is x−x1cosθ=y−y1sinθ=r$\frac{x-x_1}{\cos \theta }=\frac{y-y_1}{\sin \theta }=r$
   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)$y=\left(-\frac{A}{B}\right)x+\left(-\frac{C}{B}\right)$
   This is of the form y = m x +c, where
   m=−AB=−cofficientofxcofficientofy$m=-\frac{A}{B}=-\frac{\mathrm{cof}ficientofx}{cofficientofy}$, and intercept on y – axis =−CB=−constentcofficientofy$=-\frac{C}{B}=-\frac{constent}{cofficientofy}$
2. Transformation of Ax + By + C = 0 in intercept form xa+yb=1$\frac{x}{a}+\frac{y}{b}=1$
   x(−CA)+y(−CB)=1$\frac{x}{\left(-\frac{C}{A}\right)}+\frac{y}{\left(-\frac{C}{B}\right)}=1$
   Intercept on x – axis =−CA=−constanttermcofficientofx$=-\frac{C}{A}=-\frac{cons\tan tterm}{cofficientofx}$, Intercept on y - axis =−CB=−constanttermcofficientofy$=-\frac{C}{B}=-\frac{cons\tan tterm}{cofficientofy}$
3. Transformation of Ax + By + C = 0 in intercept form xcosα+ysinα=p$x\cos \alpha +y\sin \alpha =p$
   −AA2+B2√x−BA2+B2√y$-\frac{A}{\sqrt{A^2+B^2}}x-\frac{B}{\sqrt{A^2+B^2}}y$=CA2+B2√$=\frac{C}{\sqrt{A^2+B^2}}$
   Here cosα=−AA2+B2√$\cos \alpha =-\frac{A}{\sqrt{A^2+B^2}}$ and sinα=−BA2+B2√;p=±CA2+B2√$\sin \alpha =-\frac{B}{\sqrt{A^2+B^2}};p=\pm \frac{C}{\sqrt{A^2+B^2}}$
   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)$\left(\alpha ,\beta \right)$ 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√∣∣∣