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461.
Line equation trought two points - version 1
y−y1y−y2=x−x1x−x2 \dfrac{y-y_1}{y-y_2} = \dfrac{x-x_1}{x-x_2}
Analytic geometry
Line
462.
Line equation throught two points
∣xy1x1y11x2y21∣=0\begin{vmatrix} x&y&1\\x_1&y_1&1 \\ x_2&y_2&1 \end{vmatrix} = 0
Analytic geometry
Line
463.
Intercept form of the line equation
xa+yb=1\dfrac{x}{a} + \dfrac{y}{b} = 1
Analytic geometry
Line
464.
Normal form of a line
xcos⁡α+ysin⁡α=px\cos\alpha + y\sin\alpha = p
Analytic geometry
Line
465.
Point - direction form of a line equation
x−x1X=y−y1Y\dfrac{x-x_1}{X} = \dfrac{y-y_1}{Y}
Analytic geometry
Line
466.
Line in prametric form
x=a1+tb1y=a2+tb2\begin{aligned} x &= a_1+tb_1 \\[0.3 em] y &= a_2+tb_2 \end{aligned}
Analytic geometry
Line
Polar form
467.
Line - point distance
The distance from the point P(x0,y0)P(x_0,y_0) to the line Ax+By+C=0Ax+By+C=0 is d=∣Ax0+By0+C∣A2+B2d=\dfrac{|Ax_0 + By_0 + C|}{\sqrt{A^2+B^2}}
Analytic geometry
Distance
Line
468.
Parallel lines - verison 1
Lines y=k1x+b1y=k_1x + b_1 and y=k2x+b2y=k_2x + b_2 are parralel iff k1=k2k_1 = k_2.
Analytic geometry
Line
469.
Parallel lines - verison 2
Lines A1x+B1y+C1=0A_1x+B_1y+C_1=0 and A2x+B2y+C2=0A_2x+B_2y+C_2=0 are parallel iff A1A2=B1B2\dfrac{A_1}{A_2} = \dfrac{B_1}{B_2}.
Analytic geometry
Line
470.
Perpendicular lines - verison 1
Lines y=k1x+b1y=k_1x + b_1 and y=k2x+b2y=k_2x+b_2 are perpendicular iff k1⋅k2=1k_1\cdot k_2=1.
Analytic geometry
Line
471.
Perpendicular lines - verison 2
Lines A1x+B1y+C=0A_1x+B_1y+C=0 and A2x+B2y+C=0A_2x+B_2y+C=0 are perpendicular iff A1A2+B1B2=0A_1A_2+B_1B_2=0.
Analytic geometry
Line
472.
Angle between two lines - verison 1
tan⁡ϕ=k2−k−11+k1k2\tan\phi = \dfrac{k_2-k-1}{1+k_1k_2}
Analytic geometry
Line
473.
Angle between two lines - version 2
cos⁡ϕ=A1A2+B1B2A12+B12⋅A22+B22\cos\phi = \dfrac{A_1A_2+B_1B_2}{\sqrt{A_1^2+B_1^2} \cdot \sqrt{A_2^2+B_2^2}}
Analytic geometry
Line
474.
Intersection of two lines
Intersection point of lines A1x+B1y+C1=0A_1x+B_1y+C_1=0 and A2x+B2y+C2=0A_2x+B_2y+C_2=0 has coordinates x0=B1C2−B2C1A1B2−A2B1,y0=−A1C2+A2C1A1B2−A2B1 x_0 = \dfrac{B_1C_2 - B_2C_1}{A_1B_2-A_2B_1}, y_0 = \dfrac{-A_1C_2 + A_2C_1}{A_1B_2-A_2B_1}
Analytic geometry
Line
475.
Circle equation centered at the origin
x2+y2=R2x^2+y^2=R^2
Analytic geometry
Circle
476.
Circle equation centered at (a, b)
(x−a)2+(y−b)2=R2(x-a)^2+(y-b)^2=R^2
Analytic geometry
Circle
477.
Three point form
∣x2+y2xy1x12+y12x1y11x22+y22x2y21x32+y32x3y31∣\begin{vmatrix} x^2+y^2 & x & y & 1 \\[0.5em] x_1^2+y_1^2 & x_1 & y_1 & 1 \\[0.5em] x_2^2+y_2^2 & x_2 & y_2 & 1 \\[0.5em] x_3^2+y_3^2 & x_3 & y_3 & 1 \\[0.5em] \end{vmatrix}
Analytic geometry
Circle
478.
Parametirc equation of a circle
x=a+Rcos⁡ty=b+Rsin⁡t      0≤t≤2π\begin{aligned} x &= a + R\cos t \\ y &= b + R\sin t \\ \end{aligned} ~~~~~~ 0 \le t \le 2\pi
Analytic geometry
Circle
Polar form
479.
General circle equation
Ax2+Ay2+DX+Ey+F=0 where A≠0,D2+E2>4AFAx^2+Ay^2+DX+Ey+F=0 \text{ where } A \ne 0, D^2+E^2>4AF
Analytic geometry
Circle
480.
Ceter of a circle
If the circle equation is: Ax2+Ay2+DX+Ey+F=0Ax^2+Ay^2+DX+Ey+F=0 than the cetrer of the circle has coordinates: a=−D2A,   b=−E2Aa = -\dfrac{D}{2A}, ~~~ b = -\dfrac{E}{2A}
Analytic geometry
Circle

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