Introduction to Analytic Geometry

Introduction to Analytic Geometry

Analytic Geometry, also known as Coordinate Geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. It forms the foundation for calculus and physics.

Rectangular Coordinate System

The Cartesian Coordinate System consists of two perpendicular number lines intersecting at a point called the origin $O(0,0)$.

Distance Formula

The distance $d$ between two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ is derived from the Pythagorean Theorem.

Distance Formula

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

Midpoint Formula

The midpoint $M$ of a line segment joining two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ is the average of their coordinates.

Midpoint Formula

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

Division of a Line Segment

To find a point $P(x, y)$ that divides the segment $P_1P_2$ in the ratio $r = \frac{P_1P}{PP_2}$:

Section Formula

$x = \frac{x_1 + r x_2}{1 + r}, \quad y = \frac{y_1 + r y_2}{1 + r}$

If $r$ is positive, the point is internal. If $r$ is negative, the point is external.

Slope and Inclination

The slope $m$ of a line measures its steepness and direction. It is defined as the ratio of the "rise" (vertical change) to the "run" (horizontal change).

Slope Formula

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

The inclination $\theta$ of a line is the smallest positive angle measured counterclockwise from the positive x-axis to the line.

Relation between Slope and Inclination

$m = \tan \theta$

• If $m > 0$, the line rises to the right ($0^\circ < \theta < 90^\circ$).
• If $m < 0$, the line falls to the right ($90^\circ < \theta < 180^\circ$).
• If $m = 0$, the line is horizontal ($\theta = 0^\circ$).
• If $m$ is undefined, the line is vertical ($\theta = 90^\circ$).

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