﻿ Analytic geometry Conic section Formulas
﻿

The Parabola Formulas:

The standard formula of a parabola y2=2px Parametric equations of the parabola: \begin{aligned} x &=2\,p\,t^2 \\ y &= 2\,p\,t \end{aligned} Tangent line in a point D(x0,y0) of a parabola y2=2px is : $$y_0\,y=p\left(x+x_0\right)$$ Tangent line with a given slope m: $$y = m\,x + \frac{p}{2m}$$ Tangent lines from a given point Take a fixed point P(x0,y0). The equations of the tangent lines are: \begin{aligned} y-y_0 &= m_1(x-x_0) \\ y-y_0 &= m_2(x-x_0) \\ m_1 &= \frac{y_0 + \sqrt{y_0^2 - 2px_0}}{2x_0} \\ m_2 &= \frac{y_0 - \sqrt{y_0^2 - 2px_0}}{2x_0} \end{aligned}

The Ellipse Formulas:

The set of all points in the plane, the sum of whose distances from two fixed points, called the foci, is a constant.
The standard formula of a ellipse: $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ Parametric equations of the ellipse: \begin{aligned} x &= a\,\cos t \\ y &= b\,\sin t \end{aligned} Tangent line in a point D(x0,y0) of a ellipse: $$\frac{x_0\,x}{a^2} + \frac{y_0\,y}{b^2} = 1$$ Eccentricity of the ellipse: $$e = \frac{\sqrt{a^2-b^2}}{a}$$ Foci of the ellipse: \begin{aligned} & \text{if}~a \geq b \Longrightarrow F_1\left(-\sqrt{a^2-b^2},0\right)~~ F_2\left(\sqrt{a^2-b^2},0\right) \\ & \text{if}~a < b \Longrightarrow F_1\left(0, -\sqrt{b^2-a^2}\right) ~~ F_2\left(0, \sqrt{b^2-a^2}\right) \end{aligned} Area of the ellipse: $$A = \pi \cdot a \cdot b$$

The Hyperbola Formulas:

The set of all points in the plane, the difference of whose distances from two fixed points, called the foci, remains constant.
The standard formula of a hyperbola: $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ Parametric equations of the Hyperbola: \begin{aligned} x &= \frac{a}{\sin t} \\ y &= \frac{b\,\sin t}{\cos t} \end{aligned} Tangent line in a point D(x0,y0) of a Hyperbola: $$\frac{x_0x}{a^2} - \frac{y_0y}{b^2} = 1$$ Foci: \begin{aligned} & \text{if}~a \geq b \Longrightarrow F_1\left(-\sqrt{a^2+b^2},0\right)~~ F_2\left(\sqrt{a^2+b^2},0\right) \\ & \text{if}~a < b \Longrightarrow F_1\left(0, -\sqrt{a^2+b^2}\right) ~~ F_2\left(0, \sqrt{a^2+b^2}\right) \end{aligned} Asymptotes: \begin{aligned} & \text{if } a \geq b \Longrightarrow y = \frac{b}{a}x \text{ and } y = -\frac{b}{a}x \\ & \text{if } a < b \Longrightarrow y = \frac{a}{b}x \text{ and } y = -\frac{a}{b}x \\ \end{aligned}