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.


Slope-intercept formslope- intercept form

Two point formTwo point form

Point slope formPoint slope form

Intercept formIntercept form

Normal formNormal form

Concurrent lines Three linesconcurrent lineare concurrent if and only if:concurrent

Parametric formParametric form

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

General formgeneral form


Line segmentLine segment

satisfy 0 ≤ s ≤ 1 and 0 ≤ t ≤ 1



The area of the triangle formed by the three lines: Area of the triangle formulais given by area

The area of a triangle whose vertices are 


The centroid o f a triangle whose vertices areCENTROID


The vertices of a triangle whose incenter are



The circumcenter of a triangle whose vertices arecircumstance


The orthocenter of a triangle whose vertices areorthocenter


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:equation of a circleCircle is the centered at the regionCircle is the centered at the regionParametric equationsParametric equations

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

AreaArea equation



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.


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:parabola's standard formula

The parabola’s parametric equations:Parabola's parametric equations

Tangent lineTangent line

Tangent line with a given slope (m)Tangent line with a given slope (m)

Tangent lines from a given pointTangent line with a given point


Eccentricity:Eccentricity formula



    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
Hyperbola's standard formula

The hyperbola’s parametric equationHyperbola's parametric equation

Foci:FOCI equation



Plane forms

Point direction form: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:intercept form formulathis plane passes through the points ( a, 0,0), (0,b,0), and (0,0,c)

Three point form:three point form

Normal  form:normal form equation

Parametric form:parametric formula

Angle between two planes:

The angle between two planesangle between two planes

The planes are parallel id and only ifPlanes parallel

The planes are perpendicular if and only ifPlanes are perpendicular

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)

Equation of a plane


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.

equation of a plane through parallel



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



The intersection of two planes

A1y + B1y + C1z +D1 = 0,

A2x + B2y + C2z + D2 =0


is the line

Intersection of two planes