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Math Formulas: Conic Sections

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 $m$:

$$ y = m\,x + \frac{p}{2m} $$

Tangent lines from a given point

Take a fixed point $P(x_0, y_0)$. 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 $D(x_0, y_0)$ 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 $D(x_0, y_0)$ 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:

$$ \begin{aligned} & \text{if } a \geq b \Longrightarrow y = \frac{b}{a}x \text{ and } y = -\frac{b}{a}x \\ & \text{if } a < b \Longrightarrow y = \frac{a}{b}x \text{ and } y = -\frac{a}{b}x \\ \end{aligned} $$

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