## ANALYTIC GEOMETRY FORMULAS

Analytic Geometry is a branch of algebra, an invention of Descartes and Fermat. It deals with modelling some geometrical objects, such as lines, points, curves, and so on. The study of analytic geometry is important as it gives the knowledge for the next level of mathematics.

## LINES IN TWO DIMENSIONS

Slope-intercept form

Two point form

Point slope form

Intercept form

Normal form

Concurrent lines Three linesare concurrent if and only if:

Parametric form

Point direction formwhere (A, B) is the direction of the line and  P1 (X1,Y1) lies on the line.

General form

Distance

Line segment

satisfy 0 ≤ s ≤ 1 and 0 ≤ t ≤ 1

## TRIANGLES IN TWO DIMENSIONS

Area

The area of the triangle formed by the three lines: is given by

The area of a triangle whose vertices are

Centroid

The centroid o f a triangle whose vertices are

Incenter

The vertices of a triangle whose incenter are

Circumcenter

The circumcenter of a triangle whose vertices are

Orthocenter

The orthocenter of a triangle whose vertices are

## CIRCLE

Equation of a circle

The circle with the center (a, b) and radius r in a x-y coordinate system in the set of all points ( x, y) such that:Circle is the centered at the regionParametric equations

where is the parametric variable. In polar coordinates the equation of a circle is:

Area

Circumference

THEOREMES:

Chord theorem: The chord theorem states that if two chords, CD and EF intersect at G, then:

CD • DG = EG •FG

Tangent- secant theorem: if the tangent from an external point D intersects the circle at C and a secant from the external point D intersects the circle at G and E, then:

DC² = DG • DE

The secant-secant theorem: states that if two secants, DG & DE, cut the circle at H and F, respectively, then

DH •DG = DF • DE

Property of the tangent chord: The angle formed by a tangent and a chords is the equal to the subtended angle on the chord’s opposite side.

## CONIC SECTION

The Parabola

The set all points in the plane whose distances from a fixed point, referred to as the focus, and a fixed line, known as the directrix, are always equal.

A parabola’s standard formula is:

The parabola’s parametric equations:

Tangent line

Tangent line with a given slope (m)

Tangent lines from a given point

Eccentricity:

Foci:

Area:

K= π • a • b

The Hyperbola

All points in the plane whose distances from the two fixed points, known as foci, stay constant

The hyperbola’s standard formula

The hyperbola’s parametric equation

Foci:

Asymptotes:

## PLANES IN THREE DIMENSIONS

Plane forms

Point direction form:

where ( x1, y1, z1) lies in the plane, and the direction (a, b, c) is normal plane.

General form:

Ax + By + Cz + D = 0

where direction ( A, B,C) is the normal to the plane.

Intercept form:this plane passes through the points ( a, 0,0), (0,b,0), and (0,0,c)

Three point form:

Normal  form:

Parametric form:

Angle between two planes:

The angle between two planes

The planes are parallel id and only if

The planes are perpendicular if and only if

Equation of a plane

The equation of a plane through P1 ( X1, Y1, Z1) and parallel to directions( a1, b1, c1) and ( a2, b2, c2)

The equation of a plane through P1 (X1, Y1, Z1) and P2 (a2, b2, c2) and parallel to the direction (a, b, c), has equation.

Distance

The distance of P1 (x1, y1, z1) from the plane Ax + By + Cz + D= 0 is

Intersection

The intersection of two planes

A1y + B1y + C1z +D1 = 0,

A2x + B2y + C2z + D2 =0

is the line