Class 11 Notes - Coordinate Geometry

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Coordinate Geometry

Coordinate Geometry, also known as Analytic Geometry, is the study of geometry using a coordinate system. It allows us to represent geometric shapes algebraically and solve geometric problems using equations and coordinates.

1. Cartesian Coordinate System

  • The Cartesian plane consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical), intersecting at the origin (0, 0).
  • Any point in the plane is represented as an ordered pair (x, y).
  • The x-coordinate shows the position along the horizontal axis, and the y-coordinate shows the position along the vertical axis.
Example: The point (3, 2) is 3 units to the right of the origin and 2 units up.

2. Distance Formula

The distance between two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:

\( \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
Example: Find the distance between (1, 2) and (4, 6).
\( = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)

3. Section Formula

The section formula finds the coordinates of a point dividing the line segment joining \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \):

\( \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \)
Example: The point dividing (2, 3) and (8, 7) in the ratio 1:2 is:
\( \left( \frac{1 \times 8 + 2 \times 2}{1+2}, \frac{1 \times 7 + 2 \times 3}{1+2} \right) = (4, 13/3) \)

4. Area of a Triangle

The area of a triangle with vertices at \( (x_1, y_1), (x_2, y_2), (x_3, y_3) \) is:

\( \text{Area} = \frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)| \)
Example: Area of triangle with vertices (1, 2), (4, 6), (5, 2):
\( = \frac{1}{2} |1(6-2) + 4(2-2) + 5(2-6)| = \frac{1}{2} |4 + 0 - 20| = \frac{1}{2} \times 16 = 8 \)

5. Collinearity of Points

  • Three points are collinear if the area of the triangle formed by them is zero.
  • Alternatively, if the slopes between each pair of points are equal, the points are collinear.

6. Practice Questions

  1. Plot the points (2, 3), (4, 5), and (6, 7) on the Cartesian plane.
  2. Find the distance between the points (1, 2) and (7, 5).
  3. Find the coordinates of the point dividing the line joining (3, 4) and (9, 10) in the ratio 2:1.
  4. Check if the points (1, 2), (3, 6), and (5, 10) are collinear.
  5. Find the area of the triangle with vertices (0, 0), (4, 0), and (4, 3).

7. Summary

  • Coordinate geometry connects algebra and geometry using coordinates.
  • Distance and section formulas help solve geometric problems algebraically.
  • Area formula and collinearity are important applications.