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Coordinate geometry, or analytic geometry, uses a Cartesian coordinate system to analyze geometric figures algebraically. It covers linear equations, graphing straight lines, circle equations, and parametric equations for describing curves and motion. This field is crucial in physics, engineering, and computer graphics for spatial analysis.

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## Definition and Purpose

### Coordinate Geometry

Coordinate geometry is a mathematical discipline that uses a coordinate system to precisely analyze geometric figures

### Analytic Geometry

Analytic geometry is another name for coordinate geometry

### Purpose

The purpose of coordinate geometry is to represent geometric entities algebraically and perform calculations to determine distances, midpoints, slopes, and equations of lines and curves

## Cartesian Coordinate System

### Definition

The Cartesian coordinate system is defined by two perpendicular number lines, the x-axis (horizontal) and the y-axis (vertical)

### Components

x-axis

The x-axis is the horizontal number line in the Cartesian coordinate system

y-axis

The y-axis is the vertical number line in the Cartesian coordinate system

Origin

The origin is the point where the x-axis and y-axis intersect, dividing the plane into four quadrants

## Linear Equations

### Definition

Linear equations are algebraic expressions that form the basis of coordinate geometry and are represented graphically by straight lines on the Cartesian plane

### General Form

The general form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept

### Properties

Slope

The slope of a line measures its steepness and direction, with positive slopes indicating an upward tilt and negative slopes a downward tilt

Y-intercept

The y-intercept is the point where the line crosses the y-axis

Parallel and Perpendicular Lines

The concept of slope is instrumental in determining whether lines are parallel (having equal slopes) or perpendicular (having slopes that are negative reciprocals of each other)

## Circles

### Definition

A circle is a fundamental shape in coordinate geometry, defined as a set of points equidistant from a central point

### Standard Equation

The standard equation for a circle in the Cartesian plane is (x-h)^2 + (y-k)^2 = r^2, where (h, k) is the center of the circle and r is its radius

### Properties

Pythagorean Theorem

The equation for a circle is derived from the Pythagorean theorem

Tangency

A tangent to a circle is a line that intersects the circle at exactly one point and is perpendicular to the radius at that point

Equation of a Tangent Line

The equation of a tangent line can be found using the point of tangency and the slope, which is the negative reciprocal of the slope of the radius at that point

Algorino

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