Analytical Geometry - Equation Of Line

 
   
     
       

Coordinate Geometry – Line

       

Equation of a Line

       

          In coordinate geometry, a line can be described using various forms depending on the given points, slope, or other line characteristics.        

     
     
       
         
1. Line Joining Two Points \( A(x_1, y_1) \), \( B(x_2, y_2) \)
         

            Line through two points          

         

            \[             \frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1}             \]          

         
2. Line Through a Point and Parallel to a Given Line
         

Point: \( A(x_0, y_0) \), Given Line: \( y = ax + b \)

         

            Parallel Line Equation          

         

            \[             y - y_0 = a(x - x_0)             \]          

         
3. Line Through a Point and Perpendicular to a Given Line
         

Point: \( A(x_0, y_0) \), Given Line: \( y = ax + b \)

         

            Perpendicular Line Equation          

         

            \[             y - y_0 = -\frac{1}{a}(x - x_0)             \]          

         

Key Properties of Line Equations:

         
               
  • A line is uniquely determined by two points or a point and slope.
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  • Parallel lines have equal slopes.
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  • Perpendicular lines have slopes that are negative reciprocals.
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  • The general form of a line is: \( Ax + By + C = 0 \).
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Applications of Line Equations:

         
               
  • Finding shortest distance from a point to a line (optimization problems).
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  • Analyzing linear trends in data (regression lines).
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  • Used in graphics and game development for motion paths.
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  • Important in physics for representing velocity vs. time or force vs. displacement graphs.
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