Coordinate Geometry (Complete)

Coordinate geometry, also known as analytical geometry, is a branch of mathematics that combines algebra and geometry to study geometric shapes using algebraic methods. It involves representing geometric figures and solving geometric problems using coordinate systems and equations.

Key concepts in coordinate geometry include:

1. Coordinate System: A coordinate system is a framework used to locate points in space using numerical coordinates. The most common coordinate system is the Cartesian coordinate system, which consists of two perpendicular axes (x-axis and y-axis) intersecting at a point called the origin.

2. Coordinates: Coordinates are pairs of numbers that specify the position of a point in a coordinate system. In a Cartesian coordinate system, the coordinates of a point are given as (x, y), where x represents the horizontal distance from the y-axis (called the abscissa), and y represents the vertical distance from the x-axis (called the ordinate).

3. Distance Formula: The distance formula is used to calculate the distance between two points in a coordinate system. It is derived from the Pythagorean theorem and is given by:

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

4. Slope: Slope is a measure of the steepness of a line. It represents the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ \text{Slope} = \frac{y_2 - y_1}{x_2 - x_1} \]

Coordinate geometry is used to analyze geometric shapes, such as points, lines, circles, and curves, by representing them as equations and solving algebraic equations related to their properties. It has wide-ranging applications in fields such as physics, engineering, computer graphics, and navigation, where geometric shapes and spatial relationships play a crucial role.