# Class 11 Maths Revision Notes for Chapter-10 Straight Lines

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

**CBSE Class 11 Mathematics Revision Notes Chapter-10 Straight Lines**

**Slope of a Line****Various Forms of the Equation of a Line****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 $\left({\text{x}}_{1}\text{},\text{}{\text{y}}_{1}\text{}\right)$ and $\left({\text{x}}_{2}\text{},\text{}{\text{y}}_{2}\right)$ is given by is given by $\text{m}=\frac{{y}_{1}-{y}_{2}}{{x}_{1}-{x}_{2}}\text{=}$ ${x}_{1}\ne {x}_{2}$.

- If a line makes an angle á with the positive direction of x-axis, then the slope of the line is given by $\text{m =tan}\alpha \text{,}\alpha \ne \text{9}{\text{0}}^{o}\text{}$

- Slope of horizontal line is zero and slope of vertical line is undefined.
- An acute angle (say θ) between lines ${\text{L}}_{1}\text{and}{\text{L}}_{2}$ with slopes ${\text{m}}_{1}\text{and}{\text{m}}_{2}$ is given by $\mathrm{tan}\theta =\left|\frac{{m}_{2}-{m}_{1}}{1+{m}_{1}{m}_{2}}\right|$, $1+{m}_{1}{m}_{2}\ne 0$

- Two lines are parallel if and only if their slopes are equal i.e., ${m}_{1}={m}_{2}$
- Two lines are
*perpendicular*if and only if product of their slopes is –1, i.e., ${m}_{1}.{m}_{2}=-1$ - 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 $\left({\text{x}}_{o},\text{}{\text{y}}_{0}\text{}\right),$ 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 $\left({\text{x}}_{1},\text{}{\text{y}}_{1}\right)$ and ($\text{(}{\text{x}}_{2},\text{}{\text{y}}_{2})$ is given by $y-{y}_{1}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}(x-{x}_{1})$**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 $y=\text{mx +c}$.- If a line with slope m makes x-intercept d. Then equation of the line is $y=\text{m(x -d)}$.
**Intercept form**: Equation of a line making intercepts a and b on the x-and y-axis, respectively, is $\frac{x}{a}+\frac{y}{b}=1$.**Normal form**: The equation of the line having normal distance from origin p and angle between normal and the positive $\text{x}-\text{axis}\omega $ is given by $\text{x cos}\omega \text{+ysin}\omega =p$**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 $Ax+By=-C$

(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 $\sqrt{{\mathrm{A}}^{2}+{\mathrm{B}}^{2}}$.

The equation so obtained is in the normal form.

**Parametric Equation (Symmetric Form)**: $\frac{x-{x}_{1}}{\mathrm{cos}\theta}=\frac{y-{y}_{1}}{\mathrm{sin}\theta}=r$**Equation of a line through origin**: $y=mx$ or $y=x\mathrm{tan}\theta $.- The perpendicular distance (d) of a line Ax + By+ C = 0 from a point $\text{}\text{}\left({\text{x}}_{1},\text{}{\text{y}}_{1}\right)$ is given by $d=\frac{\left|A{x}_{1}+B{y}_{1}+C\right|}{\sqrt{{A}^{2}+{B}^{2}}}$
- Distance between the parallel lines $\text{Ax}+\text{By}+\text{}{\text{C}}_{1}$= 0 and $\text{Ax}+\text{By}+\text{}{\text{C}}_{2}$= 0, is given by $d=\frac{\left|{C}_{1}-{C}_{2}\right|}{\sqrt{{A}^{2}+{B}^{2}}}$

**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**:

${a}_{1}\left({b}_{2}{c}_{3}-{b}_{3}{c}_{2}\right)+{b}_{1}\left({c}_{2}{a}_{3}-{c}_{3}{a}_{2}\right)+{c}_{1}\left({a}_{2}{b}_{3}-{a}_{3}{b}_{2}\right)=0$

**EQUATIONS OF FAMILY OF LINES THROUGH THE INTERSECTION OF TWO LINES**

${\mathrm{A}}_{1}x+{\mathrm{B}}_{1}y+{\mathrm{C}}_{1}+k\left({\mathrm{A}}_{2}x+{\mathrm{B}}_{2}y+{\mathrm{C}}_{2}\right)=0$

where $k$ 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 = $\mathrm{tan}\theta $.

- Since a line parallel to x –axis makes an angle of 00 with x – axis, therefore its slope is tan 0° = 0.
- A line parallel to y – axis makes an angle of 90° with x – axis, so its slope is $\mathrm{tan}\frac{\pi}{2}=\mathrm{\infty}$.

**Slope of Line when Passing from two given points:**

If P(x_{1}, y_{1}) & (x_{2}, y_{2}) So, $m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$

**Angle between two Lines:**

$\mathrm{tan}\theta =\left|\frac{{m}_{2}-{m}_{1}}{1+{m}_{1}{m}_{2}}\right|$ here m1:m2 are slope of lines and $\theta $ is angle $\frac{b}{w}$ two lines.

**NOTE:** 1. If two lines are parallel to each other ⇒ m_{1} = m_{2} because $\theta =0$

- if two line are perpendicular to each other ⇒ m
_{1}m_{2}= -1 because $\theta =90$ - if line parallel to x - axis ⇒ equation of line y = k
- if line parallel to y - axis ⇒ equation of line x = k
- 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(x_{1},y_{1}): B(x_{2},y_{2}) & (x_{3},y_{3}) ⇒ slope of BC = slope of AC

**Different forms of the equation of a straight line:**

**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)**Point - slope form of a line:**

The equation of a line which passes through the point (given) P(x_{1}, y_{1}) and has the slope ‘m’ is

y - y_{1}= m(x - x_{1}).**Two point form of a line:**

The equation of a line passing through two points P(x_{1}, y_{2}) and Q(x_{2}, y_{2}) is

$y-{y}_{1}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\left(x-{x}_{1}\right)$**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 $\frac{x}{a}+\frac{y}{b}=1$.**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 $\alpha $ with x – axis is $x\mathrm{cos}\alpha +y\mathrm{sin}\alpha =p$

**Distance form of a line:**

The equation of the straight line passing through (x_{1}, y_{1}) and making an angle $\theta $ with the +ve direction of x – axis is $\frac{x-{x}_{1}}{\mathrm{cos}\theta}=\frac{y-{y}_{1}}{\mathrm{sin}\theta}=r$

Where r is the distance of the point (x, y) on the line from the point (x_{1}, y_{1})

**Transformation of general equation in different standard forms:**

- Transformation of Ax + By + C = 0 in the slope intercept form y = m x + c

$y=\left(-\frac{A}{B}\right)x+\left(-\frac{C}{B}\right)$

This is of the form y = m x +c, where

$m=-\frac{A}{B}=-\frac{\mathrm{cof}\phantom{\rule{thickmathspace}{0ex}}ficient\phantom{\rule{thickmathspace}{0ex}}of\phantom{\rule{thickmathspace}{0ex}}x}{cof\phantom{\rule{thickmathspace}{0ex}}ficient\phantom{\rule{thickmathspace}{0ex}}of\phantom{\rule{thickmathspace}{0ex}}y}$, and intercept on y – axis $=-\frac{C}{B}=-\frac{constent}{cof\phantom{\rule{thickmathspace}{0ex}}ficient\phantom{\rule{thickmathspace}{0ex}}of\phantom{\rule{thickmathspace}{0ex}}y}$ - Transformation of Ax + By + C = 0 in intercept form $\frac{x}{a}+\frac{y}{b}=1$

$\frac{x}{\left(-\frac{C}{A}\right)}+\frac{y}{\left(-\frac{C}{B}\right)}=1$

Intercept on x – axis $=-\frac{C}{A}=-\frac{cons\mathrm{tan}t\phantom{\rule{thickmathspace}{0ex}}term}{cof\phantom{\rule{thickmathspace}{0ex}}ficient\phantom{\rule{thickmathspace}{0ex}}of\phantom{\rule{thickmathspace}{0ex}}x}$, Intercept on y - axis $=-\frac{C}{B}=-\frac{cons\mathrm{tan}t\phantom{\rule{thickmathspace}{0ex}}term}{cof\phantom{\rule{thickmathspace}{0ex}}ficient\phantom{\rule{thickmathspace}{0ex}}of\phantom{\rule{thickmathspace}{0ex}}y}$ - Transformation of Ax + By + C = 0 in intercept form $x\mathrm{cos}\alpha +y\mathrm{sin}\alpha =p$

$-\frac{A}{\sqrt{{A}^{2}+{B}^{2}}}x-\frac{B}{\sqrt{{A}^{2}+{B}^{2}}}y$$=\frac{C}{\sqrt{{A}^{2}+{B}^{2}}}$

Here $\mathrm{cos}\alpha =-\frac{A}{\sqrt{{A}^{2}+{B}^{2}}}$ and $\mathrm{sin}\alpha =-\frac{B}{\sqrt{{A}^{2}+{B}^{2}}};p=\pm \frac{C}{\sqrt{{A}^{2}+{B}^{2}}}$

**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:**

**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.**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 $\left(\alpha ,\beta \right)$ to a line ax + by + c = 0 is

**Distance B/W Parallel lines:**

The distance between two parallel lines ax + by +c_{1} = 0 and ax + by + c_{2} = 0 is

$\left|\frac{{c}_{1}-{c}_{2}}{\sqrt{{a}^{2}+{b}^{2}}}\right|$