Analytic Geometry
Essential formulas and concepts in analytic geometry
Distance Between Two Points (2D)
Variables:
- \((x_1, y_1) and (x_2, y_2) are the coordinates of the two points.\)
Description/Usage:
Calculates the straight-line distance between two points in the plane, derived from the Pythagorean theorem.
Midpoint of Two Points
Variables:
- \((x_1, y_1) and (x_2, y_2) are the coordinates of the endpoints of a line segment; M is the midpoint coordinates.\)
Description/Usage:
Gives the coordinate of the point exactly in the middle of two given points.
Slope of a Line
Variables:
- \((x_1, y_1) and (x_2, y_2) are two distinct points on a line; m is the slope.\)
Description/Usage:
Measures the steepness or inclination of a line as the ratio of vertical change to horizontal change.
Equation of a Line (Slope-Intercept Form)
Variables:
- \(m is the slope of the line, b is the y-intercept (the y value when x=0), x,y are the coordinates of any point on the line.\)
Description/Usage:
Represents a line in the plane with slope \(m\) and intercept \(b\). The graph is a straight line crossing the \(y\)-axis at \((0,b)\).
Circle (Standard Equation)
Variables:
- \((h, k) is the center of the circle, r is the radius, (x, y) is any point on the circle.\)
Description/Usage:
Describes all points that are a fixed distance \(r\) from a center \((h,k)\).
Ellipse (Standard Form)
Variables:
- \((h,k) is the center of the ellipse, a is the semi-major radius (horizontal if a>b), b is the semi-minor radius (vertical if a>b).\)
Description/Usage:
Represents an ellipse – the set of points for which the sum of distances to two foci is constant. If \(a = b\), this reduces to a circle.
Equation of a Plane (3D)
Variables:
- \(A, B, C, D are constants defining the plane; (x,y,z) is any point on the plane. (The vector (A,B,C) is normal perpendicular to the plane.)\)
Description/Usage:
General form of a plane in three-dimensional space. Any point satisfying this linear equation lies on the plane.