Class 11 Mathematics: Straight Lines Notes

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1. Introduction

• A straight line is the locus of points that satisfies a linear equation in two variables.
• General form:

$$Ax + By + C = 0$$Ax+By+C=0

where $A, B, C$A,B,C are constants and $A$A and $B$B are not both zero.

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2. Slope of a Line

• Slope (m): Ratio of change in y to change in x between two points:

$$m = \frac{y_2 – y_1}{x_2 – x_1}$$m=x2​−x1​y2​−y1​​

• Parallel lines: Slopes are equal ($m_1 = m_2$m1​=m2​)
• Perpendicular lines: Product of slopes = -1 ($m_1 m_2 = -1$m1​m2​=−1)

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3. Forms of the Equation of a Line

1. Slope-Intercept Form:

$$y = mx + c$$y=mx+c

• $m$m = slope, $c$c = y-intercept

2. Point-Slope Form:

$$y – y_1 = m(x – x_1)$$y−y1​=m(x−x1​)

• Line passing through $(x_1, y_1)$(x1​,y1​) with slope $m$m

3. Two-Point Form:

$$\frac{y – y_1}{y_2 – y_1} = \frac{x – x_1}{x_2 – x_1}$$y2​−y1​y−y1​​=x2​−x1​x−x1​​

• Line passing through points $(x_1, y_1)$(x1​,y1​) and $(x_2, y_2)$(x2​,y2​)

4. Intercept Form:

$$\frac{x}{a} + \frac{y}{b} = 1$$ax​+by​=1

• $a$a = x-intercept, $b$b = y-intercept

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4. Angle Between Two Lines

• Lines with slopes $m_1$m1​ and $m_2$m2​ form angle $\theta$θ:

$$\tan \theta = \left| \frac{m_1 – m_2}{1 + m_1 m_2} \right|$$tanθ=​1+m1​m2​m1​−m2​​​

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5. Distance of a Point from a Line

• Line: $Ax + By + C = 0$Ax+By+C=0
• Point: $(x_0, y_0)$(x0​,y0​)

$$d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$$d=A2+B2​∣Ax0​+By0​+C∣​

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6. Distance Between Two Parallel Lines

• Lines: $Ax + By + C_1 = 0$Ax+By+C1​=0 and $Ax + By + C_2 = 0$Ax+By+C2​=0

$$d = \frac{|C_2 – C_1|}{\sqrt{A^2 + B^2}}$$d=A2+B2​∣C2​−C1​∣​

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7. Conditions

1. Parallel lines: $\frac{A_1}{A_2} = \frac{B_1}{B_2} \neq \frac{C_1}{C_2}$A2​A1​​=B2​B1​​=C2​C1​​
2. Perpendicular lines: $A_1 A_2 + B_1 B_2 = 0$A1​A2​+B1​B2​=0

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 Special Cases

1. Horizontal Line: y = c → slope = 0
2. Vertical Line: x = k → slope is undefined

8. Key Points

• Straight line is simplest geometric figure in coordinate geometry.
• Slope determines inclination.
• Forms of equations are interchangeable depending on known data.
• Distance formulas are useful in perpendicularity and parallelism problems.