Coordinate Geometry – Class X Mathematics

 Mathematics is not only about numbers and operations but also about the visualization of ideas. Coordinate Geometry, also known as Cartesian Geometry, provides a bridge between algebra and geometry by representing geometric figures in a plane using algebraic equations. Introduced by the French mathematician René Descartes, this branch of mathematics is a core component of the Class X curriculum and serves as the foundation for higher studies in mathematics, physics, and engineering.

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1. Basics of Coordinate Geometry

In coordinate geometry, the position of a point in a plane is described with respect to two mutually perpendicular lines:

• X-axis (horizontal axis)

• Y-axis (vertical axis)

These axes intersect at a point called the origin (0,0). The plane formed is known as the Cartesian plane.

A point $P$ in the plane is represented as an ordered pair $(x,y)$, where:

• $x$ = abscissa (distance from the y-axis),

• $y$ = ordinate (distance from the x-axis).

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2. Quadrants of the Cartesian Plane

The Cartesian plane is divided into four quadrants by the axes:

1. First Quadrant: $(+x,+y)$

2. Second Quadrant: $(-x,+y)$

3. Third Quadrant: $(-x,-y)$

4. Fourth Quadrant: $(+x,-y)$

Each quadrant indicates the sign of the coordinates.

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3. Distance Formula

The distance formula is used to calculate the distance between two points $A(x_1,y_1)$ and $B(x_2,y_2)$:

$$AB = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$

Example: Distance between $A(2,3)$ and $B(5,7)$:

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

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4. Section Formula

The section formula determines the coordinates of a point dividing a line segment internally in a given ratio.

If a point $P(x,y)$ divides the line joining $A(x_1,y_1)$ and $B(x_2,y_2)$ in the ratio $m:n$, then:

$$x = \frac{mx_2+nx_1}{m+n}, \quad y = \frac{my_2+ny_1}{m+n}$$

Example: Point dividing line joining $A(2,3)$ and $B(6,7)$ in ratio 1:1 is:

$$x = \frac{6+2}{2} = 4, \quad y = \frac{7+3}{2} = 5$$

So midpoint = $(4,5)$.

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5. Midpoint Formula

The midpoint is a special case of the section formula (ratio $1:1$):

$$M = \left(\frac{x_1+x_2}{2}, \, \frac{y_1+y_2}{2}\right)$$

Example: Midpoint of $(2,5)$ and $(6,7)$ is:

$$M = \left(\frac{2+6}{2}, \frac{5+7}{2}\right) = (4,6)$$

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6. Slope of a Line (Introductory)

The slope of a line joining two points $A(x_1,y_1)$ and $B(x_2,y_2)$ is given by:

$$m = \frac{y_2-y_1}{x_2-x_1}, \quad (x_1 \neq x_2)$$

Slope measures the steepness of the line.

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7. Applications of Coordinate Geometry

1. Geometry Problems: Distance between points, midpoints, and centroids.

2. Physics: Describing motion, the trajectory of objects.

3. Engineering: Design of structures, mapping, and computer graphics.

4. Navigation: Locating positions on maps using coordinates.

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8. Summary of Key Formulas

1. Distance Formula:

   $$AB = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$
   

2. Section Formula (Internal Division):

   $$\left(\frac{mx_2+nx_1}{m+n}, \, \frac{my_2+ny_1}{m+n}\right)$$
   

3. Midpoint Formula:

   $$\left(\frac{x_1+x_2}{2}, \, \frac{y_1+y_2}{2}\right)$$
   

4. Slope Formula:

   $$m = \frac{y_2-y_1}{x_2-x_1}$$

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9. Conclusion

Coordinate Geometry provides a powerful framework for linking algebra and geometry. By using coordinates, geometric problems become algebraic, allowing us to compute distances, midpoints, and divisions of line segments with precision. For Class X students, mastering this topic builds the foundation for advanced concepts in straight lines, circles, conic sections, calculus, and vector algebra in higher classes.

It is a subject where logic meets visualization, and numbers translate into geometry, opening doors to numerous applications in science, technology, and beyond.