# Straight Lines class 11 Notes Mathematics

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## Straight Lines class 11 Notes Mathematics

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Revision Notes for Class 11 Mathematics

## 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 $\left( {{\text{x}}_{1}}\text{ },\text{ }{{\text{y}}_{1}}\text{ } \right)$ 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 $\tan \theta =\left| \frac{{{m}_{2}}-{{m}_{1}}}{1+{{m}_{1}}{{m}_{2}}} \right|$, $1+{{m}_{1}}{{m}_{2}}\ne 0$
• 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 $\sqrt {{{\rm{A}}^2} + {{\rm{B}}^2}}$.

The equation so obtained is in the normal form.

• Parametric Equation (Symmetric Form): ${{x - {x_1}} \over {\cos \theta }} = {{y - {y_1}} \over {\sin \theta }} = r$
• Equation of a line through origin: $y=mx$ or $y = x\tan \theta$.
• The perpendicular distance (d) of a line Ax + By+ C = 0 from a point $~~\left( {{\text{x}}_{1}},~{{\text{y}}_{1}} \right)$ 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 = 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

${{\rm{A}}_1}x + {{\rm{B}}_1}y + {{\rm{C}}_1} + k\left( {{{\rm{A}}_2}x + {{\rm{B}}_2}y + {{\rm{C}}_2}} \right) = 0$

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  and $y=0$.

(ii) The perpendicular distance of this point from the other line is the required distance between the lines.

## CBSE Class-11 Revision Notes and Key Points

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