Position of Two Points with Respect to a Given Line

Introduction:

  • Understanding the position of points relative to a line is essential in coordinate geometry.
  • It involves determining whether points lie on the line, on the same side, or on different sides concerning the given line.

Point-Position Relation to a Line:

  1. On the Line:

    • A point (x, y) is on the line if it satisfies the equation of the line when substituted into the equation.
  2. Above or Below the Line:

    • Substituting a point's coordinates into the line equation determines whether the point lies above or below the line.
    • If the substituted point yields a positive value in the equation, it lies above the line; otherwise, it lies below.
  3. Side Determination using Inequality:

    • Given a line equation Ax+By+C=0, substitute the point coordinates into the equation: Ax+By+C>0 or Ax+By+C<0.
    • If the inequality is true, the point is on the respective side of the line; if false, it's on the opposite side.

Example:

Consider the line 3x4y+5=0 and two points:

  1. Point A(2, 3)
  2. Point B(1, 6)

Position of Points with Respect to the Line:

  • Point A(2, 3):

    • Substitute into the line equation: 3(2)4(3)+5=612+5=1<0
    • Conclusion: Point A is below the line.
  • Point B(1, 6):

    • Substitute into the line equation: 3(1)4(6)+5=324+5=16<0
    • Conclusion: Point B is below the line.

Importance:

  • Understanding the point-line relationship aids in geometric analysis and problem-solving involving regions, distances, or orientations in a coordinate plane.
  • This knowledge is essential in fields like computer graphics, engineering, and architecture for determining positions of objects concerning reference lines.

Summary:

  • Determining the position of points concerning a line involves substituting their coordinates into the line equation and analyzing the result (equality or inequality).
  • This analysis helps establish whether points lie on the line, above, or below it, providing crucial spatial information.