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Math formulas: Conic sections

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The Parabola Formulas

The standard formula of a parabola

y2=2px y^2 = 2\,p\,x

Parametric equations of the parabola:

x=2pt2y=2pt \begin{aligned} x &=2\,p\,t^2 \\ y &= 2\,p\,t \end{aligned}

Tangent line in a point D(x0,y0)D(x_0, y_0) of a parabola y2=2pxy^2 = 2px is :

y0y=p(x+x0) y_0\,y=p\left(x+x_0\right)

Tangent line with a given slope mm:

y=mx+p2m y = m\,x + \frac{p}{2m}

Tangent lines from a given point

Take a fixed point P(x0,y0)P(x_0, y_0). The equations of the tangent lines are:

yy0=m1(xx0)yy0=m2(xx0)m1=y0+y022px02x0m2=y0y022px02x0 \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:

x2a2+y2b2=1 \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

Parametric equations of the ellipse:

x=acosty=bsint \begin{aligned} x &= a\,\cos t \\ y &= b\,\sin t \end{aligned}

Tangent line in a point D(x0,y0)D(x_0, y_0) of a ellipse:

x0xa2+y0yb2=1 \frac{x_0\,x}{a^2} + \frac{y_0\,y}{b^2} = 1

Eccentricity of the ellipse:

e=a2b2a e = \frac{\sqrt{a^2-b^2}}{a}

Foci of the ellipse:

if abF1(a2b2,0)  F2(a2b2,0)if a<bF1(0,b2a2)  F2(0,b2a2) \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=πab 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:

x2a2y2b2=1 \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

Parametric equations of the Hyperbola:

x=asinty=bsintcost \begin{aligned} x &= \frac{a}{\sin t} \\ y &= \frac{b\,\sin t}{\cos t} \end{aligned}

Tangent line in a point D(x0,y0)D(x_0, y_0) of a Hyperbola:

x0xa2y0yb2=1 \frac{x_0x}{a^2} - \frac{y_0y}{b^2} = 1

Foci:

if abF1(a2+b2,0)  F2(a2+b2,0)if a<bF1(0,a2+b2)  F2(0,a2+b2) \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:

if aby=bax and y=baxif a<by=abx and y=abx \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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