Cheatography
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Formulas for the basic topics of analytical geometry.
Line
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General equation Ax+By+C=0
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Point slope equation y-y1=m(x-x1)
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Slope of a straight line (m) m=-A/B
m=(y2-y1)/(x2-x1)
m=tg(α)
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Positions relative to two lines
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Equations Ax+By+C=0
A'x+B'y+C'=0
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Secant lines A/A'≠B/B'
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Parallel lines A/A'=B/B'≠ C/C'
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Coincident lines A/A'=B/B'= C/C'
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Parallel lines
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Slope m1=m2
-A/B=-A'/B'
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Perpendicular lines
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Slope m1.m2=-1
-A/B=B'/A'
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Points
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Coordinates P(x1,y1)
Q(x2,y2)
M(x3,y3)
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Distance between two points (P and Q) d=√[(x2-x1)^2+(y2-y1)^2]
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Midpoint (M) x3=(x1+x2)/2
y3=(y1+y2)/2
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Distance from a point (P) to a line d=|(A.x1+B.y1+C)/√(a^2+b^2)|
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Circumference
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Ordinary equation (x-a)^2+(y-b)^2=r^2
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Elements Center: C(a,b)
Point: P(x,y)
Radius: r
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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
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Elements A=-2a
B=-2b
C=a^2+b^2-r
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Ellipse
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Equation [(x-x0)^2]/(a^2)+[(y-y0)^2]/(b^2)=1
Center: C(x0,y0)
Horizontal radius: a
Vertical radius: b
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Parabola
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General equation Ax^2+Bxy+Cy^2+Dx+Ey+F=0
If:
A≠0 v B≠0
B^2-4AC=0
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Vertical Parabola Ordinary Equation (x-x0)^2=2p(y-y0)
Vertex: V(x0,y0)
Parameter: p
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Horizontal Parabola Ordinary Equation (y-y0)^2=2p(x-x0)
Vertex: V(x0,y0)
Parameter: p
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Hyperbola
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General equation Ax^2+Bxy+Cy^2+Dx+Ey+F=0
If:
A≠0 ^ B≠0
A.B<0
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Ordinary Equation (horizontal focal axis) [(x-x0)^2]/(a^2)-[(y-y0)^2]/(b^2)=1
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Ordinary Equation (vertical focal axis) [(y-y0)^2]/(b^2)-[(x-x0)^2]/(a^2)=1
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Equilateral Hyperbola Equation a=b
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Asymptotes y=x
y=-x
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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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