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
TRIANGLES IN TWO DIMENSIONS
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 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 with a given slope (m)
Tangent lines from a given point
PLANES IN THREE DIMENSIONS
where ( x1, y1, z1) lies in the plane, and the direction (a, b, c) is normal plane.
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)
Angle between two planes:
The planes are parallel id 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.
The distance of P1 (x1, y1, z1) from the plane Ax + By + Cz + D= 0 is
The intersection of two planes
A1y + B1y + C1z +D1 = 0,
A2x + B2y + C2z + D2 =0