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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: […]

Useful Limits Formulas

The General Limit Formulas \(\text{If } \lim_{x \to a} f(x) = l \text{ and } \lim_{x \to a} g(x) = m \text{ ,then}\) \( \lim_{x \to a} ~\left[ f(x) \pm g(x) \right] = l \pm m \) \( \lim_{x \to a} ~\left[ f(x) \cdot g(x) \right] = l \cdot m \) \( \lim_{x \to a} \frac{f(x)}{g(x)} = […]

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 […]

Conic Sections Formulas

The Parabola Formulas The standard formula of a parabola \( y^2 = 2\,p\,x \) Parametric equations of the parabola: \( \begin{aligned} x &=2\,p\,t^2 \\ y &= 2\,p\,t \end{aligned} \) Tangent line in a point \(D(x_0, y_0)\) of a parabola \(y^2 = 2px\) is: \( y_0\,y=p\left(x+x_0\right) \) Tangent line with a given slope m: \( y = m\,x + \frac{p}{2m} \) […]

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