Angle between Pair of Lines

Angle between Pair of Lines:

1. Equation of Pair of Lines:

  • A pair of lines represented by ax2+2hxy+by2+2gx+2fy+c=0 can be expressed as A1x2+2B1xy+C1y2+2D1x+2E1y+F1=0 and A2x2+2B2xy+C2y2+2D2x+2E2y+F2=0.

2. Angle between Pair of Lines Formula:

  • The formula to find the angle between two lines ax2+2hxy+by2+2gx+2fy+c=0 and px2+2qxy+ry2+2sx+2ty+u=0 is given by: tanθ=2(hpragq)(agqbps)(apr+bqs)

 where:

    • a,b,h are coefficients of the first line
    • p,r,q are coefficients of the second line
    • g,f,s,t are other coefficients

Example:

Given two lines: 3x2+4xy4y2+7x6y8=0 and 2x25xyy2+3x+2y4=0.

Find the angle between these lines.

Solution:

Comparing the given equations with the standard form of Ax2+2Bxy+Cy2+2Dx+2Ey+F=0:

  • For the first equation: A1=3, B1=2, C1=4, D1=7, E1=6
  • For the second equation: A2=2, B2=5, C2=1, D2=3, E2=2

Applying the formula for the angle between lines: tanθ=2(hpragq)(agqbps)(apr+bqs) tanθ=2((2)(2)(4)(3)(5)(7))((4)(5)(2)(3)(2))(322+(4)(5)) tanθ=2(16+105)(2012)(12+20)

tanθ=2121(32)32

tanθ=22432

tanθ=114

Taking inverse tangent to find the angle: θ=tan1(114) θ70.88

Therefore, the angle between the given lines is approximately 70.88.

3. Derivation of the Angle Formula:

  • Obtained from the formula for the angle between two lines using the slope of the lines and the formula for the angle between two vectors.

4. Properties and Observations:

  • The angle between the lines is invariant, irrespective of the way the lines are represented (e.g., in standard form or slope-intercept form).
  • If two lines are perpendicular, then the angle between them is 90 degrees, and the product of their slopes is -1.

5. Special Cases:

  • Parallel Lines: If two lines are parallel, the angle between them is either 0 degrees or 180 degrees (cosine of the angle will be 1 or -1).
  • Perpendicular Lines: When two lines are perpendicular, the angle between them is 90 degrees.

6. Applications:

  • Geometry: Useful in determining the orientation of lines and their intersections.
  • Trigonometry and Mathematics: Understanding angles between vectors and lines.
  • Physics and Engineering: Used in various calculations involving lines and their relationships.

7. Graphical Representations:

  • The concept of the angle between lines can be visually understood by plotting lines on the Cartesian plane and observing their intersection angles.