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Geometry I Cheat Sheet by

Formulas for the basic topics of analytical geometry.

Line

General equation
Ax+By+C=0
Point slope equation
y-y1=m­(x-x1)
Slope of a straight line (m)
m=-A/B
m=(y2-­y1)­/(x­2-x1)
m=tg(α)

Positions relative to two lines

Equations
Ax+By+C=0
A'x+B'y+C'=0
Secant lines
A/A'≠B/B'
Parallel lines
A/A'=B/B'≠ C/C'
Coincident lines
A/A'=B/B'= C/C'

Parallel lines

Slope
m1=m2
-A/B=-­­A'/B'

Perpen­dicular lines

Slope
m1.m2=-1
-A/B=B'/A'
 

Points

Coordi­nates
P(x1,y1)
Q(x2,y2)
M(x3,y3)
Distance between two points (P and Q)
d=[(x2-x­1)^­2+(­y2-­y1)^2]
Midpoint (M)
x3=(x1­+x2)/2
y3=(y1­+y2)/2
Distance from a point (P) to a line
d=|(A.x­1+­B.y­1+C)/(a^2+b^2)|

Circum­ference

Ordinary equation
(x-a)^­2+(­y-b­)^2=r^2
Elements
Center: C(a,b)
Point: P(x,y)
Radius: r
General equation
x^2+y^2+Ax+By+C=0
If:
(A/2)^2+(B/2)^2-C>0
x
and y don't multiply
x^2 and y^2 have 1 as coefficient
Elements
A=-2a
B=-2b
C=a^2+b^2-r

Ellipse

Equation
[(x-x0­)^2­]/(­a^2­)+[­(y-­y0)­^2]­/(b­^2)=1
Center: C(x0,y0)
Horizontal radius: a
Vertical radius: b
 

Parabola

General equation
Ax^2+B­xy+­Cy^­2+D­x+E­y+F=0
If:
A≠0 v B≠0
B^2-4AC=0
Vertical Parabola Ordinary Equation
(x-x0)­^2=­2p(­y-y0)
Vertex: V(x0,y0)
Parameter: p
Horizontal Parabola Ordinary Equation
(y-y0)­^2=­2p(­x-x0)
Vertex: V(x0,y0)
Parameter: p

Hyperbola

General equation
Ax^2+B­xy+­Cy^­2+D­x+E­y+F=0
If:
A≠0 ^ B≠0
A.B<0
Ordinary Equation (horiz­ontal focal axis)
[(x-x0­)^2­]/(­a^2­)-[­(y-­y0)­^2]­/(b­^2)=1
Ordinary Equation (vertical focal axis)
[(y-y0­)^2­]/(­b^2­)-[­(x-­x0)­^2]­/(a­^2)=1
Equila­teral Hyperbola Equation
a=b
Asymptotes
y=x
y=-x
Elements
Center: O(x0,y0)
Length of the semimajor axis of the hyperbola: a
Length of the semi-minor axis of the hyperbola: b
   
 

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