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Circle Formulas

Equation of a circle In an x−y coordinate system, the circle with center (a,b) and radius r is the set of all points (x,y) such that: \( (x-a)^2 + (y-b)^2 =r^2 \) Circle centered at the origin: \( x^2 + y^2 = r^2 \) Parametric equations \( \begin{aligned} x &= a + r\,\cos t \\ y&= b + […]

Triangles in Two Dimensions

Area of the triangle The area of the triangle formed by the three lines: \( \begin{aligned} A_1x + B_1y + C_1 &= 0 \\ A_2x + B_2y + C_2 &= 0 \\ A_3x + B_3y + C_3 &= 0 \end{aligned} \) is given by \( A = \frac{\begin{vmatrix} A_1 & B_1 & C_1 \\ A_2 […]

Lines in Two Dimensions

Line forms Slope y-intercept form: \( y = mx+b \) Two point form: \( y – y_1 =\frac{y_2-y_1}{x_2 – x_1} (x – x_1) \) Point slope form: \( y – y_1 = m(x – x_1) \) Intercept form: \( \frac{x}{a} + \frac{y}{b} = 1~,~(a,b \ne 0) \) Normal form: \( x\cdot \cos\Theta + y\cdot \sin\Theta = p \) Parametric form: […]

Planes in Three Dimensions

Plane forms Point direction form: \( a(x-x_1) + b(y-y_1) + c(z-z_1) = 0 \) where \(P(x_1, y_1, z_1)\) lies in the plane, and the direction (a,b,c) is normal to the plane. General form: \( Ax + By + Cz + D = 0 \) where direction (A,B,C) is normal to the plane. Intercept form: \( \frac{x}{a} + \frac{y}{b} + […]

Lines in Three Dimensions

Line forms Point direction form: \( \frac{x-x_1}{a} = \frac{y – y_1}{b} = \frac{z-z_1}{c} \) Two point form: \( \frac{x-x_1}{x_2-x_1} = \frac{y – y_1}{y_2-y_1} = \frac{z-z_1}{z_2-z_1} \) Parametric form: \( \begin{aligned} x &= x_1 +t\,\cos \alpha \\ y &= y_1 +t\,\cos \beta \\ z &= z_1 +t\,\cos \gamma \end{aligned} \) Distance between two lines in 3 dimensions The […]

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