# 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 $3x-4y+5=0$ 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: $3\left(2\right)-4\left(3\right)+5=6-12+5=-1<0$
• Conclusion: Point A is below the line.
• Point B(1, 6):

• Substitute into the line equation: $3\left(1\right)-4\left(6\right)+5=3-24+5=-16<0$
• 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.