Coordinate Geometry

1. Introduction to Coordinate Geometry

• Coordinate Axes: To locate a point on a plane, a pair of perpendicular axes is required.
  • The distance of a point from the y-axis is called its x-coordinate or abscissa.
  • The distance of a point from the x-axis is called its y-coordinate or ordinate.
• Point Formats:
  • Coordinates of a point on the x-axis are of the form (x, 0).
  • Coordinates of a point on the y-axis are of the form (0, y).
• Algebra and Geometry Connection: Coordinate geometry serves as an algebraic tool for studying geometry. It allows the study of geometry using algebra and the understanding of algebra through geometry.
  • Linear equations (ax + by + c = 0) graphically represent straight lines.
  • Quadratic equations (y = ax² + bx + c) graphically represent parabolas.

2. Distance Formula

This formula is used to find the distance between any two given points on a plane.

• Distance from Origin: The distance of a point P(x, y) from the origin O(0, 0) is simply:
  OP = √(x² + y²)
• Properties: Since distance is a length, the result is always non-negative. We take only the positive square root.
• Applications of Distance Formula:
  • Determining Triangles: By calculating the lengths of three sides, one can determine if a triangle is right-angled (using Pythagoras theorem), isosceles (two sides equal), or equilateral (all sides equal).
  • Collinearity: Three points A, B, and C are collinear (lie on the same line) if the sum of two distances equals the third (e.g., AB + BC = AC).
  • Quadrilaterals: It can verify shapes like squares (all sides equal, diagonals equal) or parallelograms (opposite sides equal).
  • Equidistance: It helps find a point that is at an equal distance from two other points (finding the perpendicular bisector).

3. Section Formula

This formula is used to find the coordinates of a point that divides a line segment joining two points in a specific ratio.

• Alternative Method (k : 1 Ratio):
  If the ratio is unknown, it is often easier to assume the ratio is k : 1. The coordinates are then:
  [ (kx₂ + x₁)/(k + 1) , (ky₂ + y₁)/(k + 1) ]
• Mid-Point Formula (Special Case):
  The mid-point of a line segment divides it in the ratio 1 : 1. The coordinates of the mid-point P of the segment joining A(x₁, y₁) and B(x₂, y₂) are:
  ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )
• Applications of Section Formula:
  • Trisection: Finding points that divide a line segment into three equal parts (ratios 1:2 and 2:1).
  • Parallelograms: Finding a missing vertex of a parallelogram by using the property that diagonals bisect each other (i.e., they share the same mid-point).
  • Finding Ratios: Determining the ratio in which a known point (or an axis) divides a line segment.

4. Chapter Summary Points

• Distance Formula: √[ (x₂ - x₁)² + (y₂ - y₁)² ]
• Distance from Origin: √(x² + y²)
• Section Formula: Finds the point dividing a segment in ratio m₁:m₂.
• Mid-point Formula: Finds the center point between two coordinates.
• Geometric Verification: These algebraic tools allow for the precise verification of geometric figures (squares, rectangles, triangles) and properties (collinearity, parallelism) without physical measurement.