Coordinate Geometry

Coordinate Geometry Notes

7.1 Introduction

Coordinate geometry combines algebra and geometry using the coordinate plane. Points can be located using coordinates in pairs (x, y):

• The x-coordinate (or abscissa) describes a point's horizontal distance from the y-axis.
• The y-coordinate (or ordinate) indicates a point's vertical distance from the x-axis.

In coordinate geometry, linear equations represent straight lines. For instance, the equation of the form ax + by + c = 0 represents a straight line when graphed. Quadratic equations give parabolas. This intersection of algebra and geometry is useful in various disciplines, including physics and engineering.

7.2 Distance Formula

To find the distance between points A(x1, y1) and B(x2, y2), we apply the Pythagorean theorem:

• Draw right triangles or use coordinates to reconstruct points' positions on the plane.

• The distance formula is derived as:

  [ PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

• This formula works irrespective of the quadrants in which the points reside, allowing consistent and easy distance calculations across all coordinate plane sections.

Special Cases

1. The distance of a point from the origin (0, 0) can be calculated as: [ OP = \sqrt{x^2 + y^2} ]
2. The formula can easily be used to determine if points are collinear by checking if the distance among them satisfies the triangle inequality.

Examples in the section illustrate how to:

• Determine if three points form a triangle.
• Verify if the points represent specific geometric shapes like squares or right triangles.

Example Explaining the Triangle Formed by Points

• For the points (3, 2), (-2, -3), and (2, 3), the distance between each pair is calculated using the distance formula. If the sum of the distances of any two sides is greater than the length of the third side, they form a triangle.

7.3 Section Formula

The section formula helps to find coordinates of a point P that divides AB in the ratio m:n internally. The coordinates P(x, y) are given by: [ P(x, y) = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right) ]

• If point P is external: [ P(x, y) = \left( \frac{m x_2 - n x_1}{m-n}, \frac{m y_2 - n y_1}{m-n} \right) ]
• Special cases include finding midpoints by setting m = n = 1. The midpoint formula is: [ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ]

Examples Using Section Formula

• To divide a segment joining (4, -3) and (8, 5) in the ratio 3:1, we would apply the section formula, resulting in point (7, 3) which lies along the segment.
• Finding points of trisection involves using the section formula with specific ratios to determine necessary coordinates.

These formulas are essential for solving geometrical problems involving distances and divisions in coordinate geometry.

7.4 Summary

In this chapter, key concepts include:

1. Distance Formula: Determines the distance between two points as ( PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} )
2. Distance from Origin: Distance of point P(x, y) from the origin is ( OP = \sqrt{x^2 + y^2} )
3. Section Formula: Coordinates that internally divide AB in ratio m:n are: [ P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) ]
4. Mid-point formula: For points P and Q is given as ( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ]

Use these foundational tools in coordinate geometry for calculations, problem-solving, and further mathematical studies.

1. Coordinates: Each point on the plane has an x-coordinate (abscissa) and y-coordinate (ordinate).
2. Distance Formula: The distance between points (x1, y1) and (x2, y2) is calculated using ( PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ).
3. Collinearity: Points are collinear if the sum of two distances is equal to the third distance.
4. Section Formula: The coordinates of point P dividing AB in the ratio m:n are given by ( P = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) ).
5. Midpoint Formula: The midpoint of line segment joining P and Q can be found using ( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) ).
6. Distance from Origin: The distance from the origin to a point (x, y) is ( OP = \sqrt{x^2 + y^2} ).
7. Triangles: The distance formula can validate the type and formation of triangles from coordinate points.