Coordinate Geometry Class 10 ||Maths|| Chapter 7 NCERT Notes

Introduction to the Coordinate Plane:

The coordinate plane is a two-dimensional plane formed by the intersection of two perpendicular lines:

The X-axis (horizontal line).
The Y-axis (vertical line).

These axes divide the plane into four quadrants:

Quadrant I: Both x and y are positive.
Quadrant II: x is negative, y is positive.
Quadrant III: Both x and y are negative.
Quadrant IV: x is positive, y is negative.

Each point in this plane is defined by an ordered pair (x, y), called coordinates. The first number is the x-coordinate (or abscissa), and the second number is the y-coordinate (or ordinate).

Important Formulas in Coordinate Geometry:

1. Distance Formula:

The distance between two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ in a coordinate plane is given by the formula:

Distance = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}

This formula is derived from the Pythagoras Theorem in a right-angled triangle.

Application Example: Find the distance between the points $A (2, 3)$ and $B (5, 7)$ .

Using the formula:

Distance = \sqrt{(5 - 2)^{2} + (7 - 3)^{2}} = \sqrt{9 + 16} = \sqrt{25} = 5

2. Section Formula:

The section formula helps us find the coordinates of a point that divides a line segment joining two points in a given ratio.

For internal division: If the point $P (x, y)$ divides the line segment joining two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ in the ratio $m_{1} : m_{2}$ , then the coordinates of $P$ are:

P (x, y) = (\frac{m_{1} x_{2} + m_{2} x_{1}}{m_{1} + m_{2}}, \frac{m_{1} y_{2} + m_{2} y_{1}}{m_{1} + m_{2}})

For external division: The formula remains the same, but the sign between $m_{1}$ and $m_{2}$ will change.

Application Example: Find the point that divides the line segment joining points $A (2, - 3)$ and $B (6, 1)$ in the ratio 2:1 internally.

Using the section formula:

P (x, y) = (\frac{2 (6) + 1 (2)}{2 + 1}, \frac{2 (1) + 1 (- 3)}{2 + 1})

P (x, y) = (\frac{12 + 2}{3}, \frac{2 - 3}{3}) = (\frac{14}{3}, \frac{- 1}{3})

Thus, $P (\frac{14}{3}, \frac{- 1}{3})$ is the required point.

3. Midpoint Formula:

The midpoint of a line segment joining two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ can be found using the midpoint formula:

Midpoint M (x, y) = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})

Application Example: Find the midpoint of the line segment joining $A (1, 4)$ and $B (5, 6)$ .

Using the formula:

M (x, y) = (\frac{1 + 5}{2}, \frac{4 + 6}{2}) = (3, 5)

Thus, the midpoint is $M (3, 5)$ .

4. Area of a Triangle:

The area of a triangle with vertices at points $(x_{1}, y_{1})$ , $(x_{2}, y_{2})$ , and $(x_{3}, y_{3})$ is given by the formula:

Area of triangle = \frac{1}{2} ∣ x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) ∣

The absolute value ensures that the area is positive, regardless of the order of the points.

Application Example: Find the area of the triangle with vertices $A (2, 3)$ , $B (4, 7)$ , and $C (6, 3)$ .

Using the formula:

Area = \frac{1}{2} ∣ 2 (7 - 3) + 4 (3 - 3) + 6 (3 - 7) ∣

= \frac{1}{2} ∣ 2 (4) + 4 (0) + 6 (- 4) ∣ = \frac{1}{2} ∣ 8 + 0 - 24 ∣ = \frac{1}{2} \times 16 = 8 square units

Thus, the area of the triangle is 8 square units.

Important Concepts to Remember:

Collinearity of Three Points:
If the area of a triangle formed by three points is zero, then the points are said to be collinear, i.e., they lie on the same straight line.
Slope of a Line:
Though not a part of the main chapter in Class 10, it's helpful to know that the slope of a line joining two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is:
$Slope (m) = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$
Quadrants Overview:
Each quadrant is determined by the sign of x and y coordinates:
- Quadrant I: (+, +)
- Quadrant II: (-, +)
- Quadrant III: (-, -)
- Quadrant IV: (+, -)

Key Takeaways:

Distance Formula helps to find the distance between two points in a plane.
Section Formula and Midpoint Formula help in dividing a line segment in a given ratio or finding the midpoint of the segment.
Area of a Triangle Formula is useful to determine the area when the vertices are known in coordinate form.
Quick-thinking Tip: If you ever need to check the collinearity of points, calculate the area of the triangle. If it’s zero, they are collinear.

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