49  Straight Lines

49.1 Introduction

The straight line is the simplest and most fundamental geometric object in coordinate geometry.
It is defined as the shortest distance between two points and extends infinitely in both directions.

In the Cartesian plane, a line can be represented by various forms of equations depending on the given data (slope, intercepts, points).
This chapter explores different forms of line equations, slope properties, parallelism, perpendicularity, distance formulas, angle between lines, and applications.

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49.2 Concept of Slope

The slope (m) of a line measures its inclination with respect to the positive x-axis.
For line through \(P(x_1,y_1)\) and \(Q(x_2,y_2)\): \[ m = \frac{y_2-y_1}{x_2-x_1}, \quad x_1\neq x_2 \]

• \(m=0\): horizontal line.
  
• \(m=\infty\): vertical line.
  
• Positive slope → line rises left to right.
  
• Negative slope → line falls left to right.

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49.3 Various Forms of Equation of a Line

49.3.1 1. Slope-Intercept Form

Equation: \[ y = mx + c \] - \(m\): slope
- \(c\): y-intercept (point where line meets y-axis)

Example: Line with slope \(2\), intercept \(-3\) → \(y=2x-3\).

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49.3.2 2. Point-Slope Form

For line through \((x_1,y_1)\) with slope \(m\): \[ y-y_1=m(x-x_1) \]

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49.3.3 3. Two-Point Form

Line through \((x_1,y_1)\) and \((x_2,y_2)\): \[ y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1) \]

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49.3.4 4. Intercept Form

If line cuts x-axis at \(a\) and y-axis at \(b\): \[ \frac{x}{a}+\frac{y}{b}=1 \]

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49.3.5 5. General Form

Any line can be written as: \[ ax+by+c=0 \] where \(a,b,c\) are real constants, not both \(a\) and \(b\) zero.

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49.4 Conditions of Parallelism and Perpendicularity

• Two lines \(y=m_1x+c_1\) and \(y=m_2x+c_2\):
  • Parallel if \(m_1=m_2\).
    
  • Perpendicular if \(m_1m_2=-1\).
• General form \(a_1x+b_1y+c_1=0\), \(a_2x+b_2y+c_2=0\):
  • Parallel if \(\tfrac{a_1}{a_2}=\tfrac{b_1}{b_2}\).
    
  • Perpendicular if \(a_1a_2+b_1b_2=0\).

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49.5 Angle Between Two Lines

If slopes are \(m_1,m_2\):
\[ \tan \theta = \Bigg|\frac{m_1-m_2}{1+m_1m_2}\Bigg| \]

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49.6 Distance Between a Point and a Line

For point \(P(x_1,y_1)\) and line \(ax+by+c=0\):
\[ d=\frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}} \]

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49.7 Distance Between Two Parallel Lines

For \(ax+by+c_1=0\) and \(ax+by+c_2=0\):
\[ d=\frac{|c_1-c_2|}{\sqrt{a^2+b^2}} \]

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49.8 Family of Lines

1. Passing through a point: \((y-y_1)=m(x-x_1)\) for varying \(m\).
   
2. Concurrent lines: lines intersecting at a common point.
   
3. Angle bisectors: Equations of bisectors of two lines \(a_1x+b_1y+c_1=0\) and \(a_2x+b_2y+c_2=0\) are:
   \[ \frac{a_1x+b_1y+c_1}{\sqrt{a_1^2+b_1^2}} = \pm \frac{a_2x+b_2y+c_2}{\sqrt{a_2^2+b_2^2}} \]

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49.9 Solved Examples

Example 1
Find equation of line through \((2,3)\) with slope \(4\).
Using point-slope form: \(y-3=4(x-2)\) → \(y=4x-5\).

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Example 2
Find intercept form of line cutting intercepts 3 on x-axis and 4 on y-axis.
Equation: \(x/3+y/4=1\).

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Example 3
Find slope and intercepts of line \(2x+3y-6=0\).
Slope \(m=-2/3\), x-intercept=3, y-intercept=2.

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Example 4
Find angle between lines \(y=2x+1\) and \(y=-\tfrac{1}{2}x+3\).
Slopes \(m_1=2\), \(m_2=-1/2\).
\(\tan \theta=\big|\tfrac{2-(-1/2)}{1+2(-1/2)}\big|=\big|\tfrac{2.5}{0}\big|=\infty\).
So angle = \(90^\circ\) (perpendicular).

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Example 5
Find distance of point \((3,4)\) from line \(3x-4y+5=0\).
\(d=\frac{|3(3)-4(4)+5|}{\sqrt{3^2+(-4)^2}}=\frac{|9-16+5|}{5}= \tfrac{2}{5}\).

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Example 6
Find distance between parallel lines \(2x+3y+4=0\) and \(2x+3y-6=0\).
\(d=\tfrac{|(-6)-4|}{\sqrt{2^2+3^2}}=\tfrac{10}{\sqrt{13}}\).

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49.10 Practice Problems

1. Find equation of line through \((1,2)\) and \((3,6)\).
   
2. Write slope-intercept form of \(3x-2y+4=0\).
   
3. Find equation of line parallel to \(5x-7y+9=0\) and passing through \((2,-3)\).
   
4. Find angle between \(y=x\) and \(y=-2x\).
   
5. Find perpendicular distance from \((2,5)\) to line \(x-y+1=0\).
   
6. Find equation of line with intercepts \(-4\) on x-axis and \(6\) on y-axis.
   
7. Find equation of line bisecting angle between \(x+y=0\) and \(x-y=0\).

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49.11 Summary

• Slope = measure of line’s inclination.
  
• Forms of line: slope-intercept, point-slope, two-point, intercept, general.
  
• Parallelism: equal slopes; perpendicularity: product of slopes = -1.
  
• Distance formulas are key in coordinate geometry.
  
• Families of lines and angle bisectors extend applications.
  
• Straight lines link algebra with geometry, forming the foundation for conics and calculus.