Conic Sections.md

Definition

Conic sections are graphs that are obtained by slicing a cone (with an inverted cone on top) with a plane

A-Level Further Pure 1

Eccentricity

For all points $P$ on a conic section, the ratio of the distance $P$ from a fixed point $F$ (the focus) and fixed straight line $D$ (the directrix) is constant. This ratio $e$ is called the eccentricity.

• If $0 \leq e < 1$, $P$ describes an ellipse
• If $e = 1$, $P$ describes a parabola
• If $e > 1$, $P$ describes a hyperbola

Summary of Conics and Properties

Property	Ellipse	Parabola	Hyperbola	Rectangular Hyperbola
Standard Form	$\displaystyle\frac{x^2}{a^2} + \displaystyle\frac{y^2}{b^2} = 1$	$y^2 = 4ax$	$\displaystyle\frac{x^2}{a^2} - \displaystyle\frac{y^2}{b^2} = 1$	$xy = c^2$
Parametric Form	$(a\cos\theta, b\sin\theta)$	$(at^2, 2at)$	$(a\sec\theta, b\tan\theta)$	$\left(ct, \displaystyle\frac{c}{t}\right)$
Eccentricity	$e < 1$ $b^2 = a^2(1-e^2)$	$e = 1$	$e > 1$ $b^2 = a^2(e^2-1)$	$e = \sqrt{2}$
Foci	$(\pm ae, 0)$	$(a, 0)$	$(\pm ae, 0)$	$(\pm\sqrt{2}c, \pm\sqrt{2}c)$
Directrices	$x = \pm\displaystyle\frac{a}{e}$	$x = -a$	$x = \pm\displaystyle\frac{a}{e}$	$x \pm y = \pm\sqrt{2}c$
Asymptotes	none	none	$\displaystyle\frac{y}{b} = \pm\displaystyle\frac{x}{a}$	$x = 0, y = 0$

Parabolas

Formed by slicing a cone with a plane parallel to its slant Cartesian Equation: $y^2 = 4ax$ Parametric Equation: $x = at^2$, $y = 2at$ for $t \in \mathbb{R}$ Eccentricity: $e = 1$ Focus: $(a, 0)$ Directrix: $x + a = 0$

Hyperbolas

Formed by slicing a cone with a plane that intersects both nappes of the cone Cartesian Equation: $\displaystyle\frac{x^2}{a^2} - \displaystyle\frac{y^2}{b^2} = 1$ Parametric Equation: $x = a\sec\theta$, $y = b\tan\theta$ or $x = a\cosh t$, $y = b\sinh t$ Eccentricity: $e > 1$ Focus: $(\pm ae, 0)$ Directrix: $x = \pm\displaystyle\frac{a}{e}$

Other Properties $c^2 = a^2e^2 = a^2 + b^2$ where $c$ is the focal length meaning the distance from the centre to a focus

Rectangular Hyperbolas

Formed by slicing a cone with a plane perpendicular to the axis of symmetry Cartesian Equation: $xy = c^2$ Parametric Equation: $(ct, \displaystyle\frac{c}{t})$ Eccentricity: $e = \sqrt{2}$ Focus: $(\pm\sqrt{2}c, \pm\sqrt{2}c)$ Directrix: $x \pm y = \pm\sqrt{2}c$

Ellipses

For major axis on the $x$-axis meaning $a > b$ - If the major axis is the y-axis so $b>a$ swap a and b around for everything Formed by slicing a cone with a plane at an angle less than that of the cone's slant Cartesian Equation: $\displaystyle\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ Parametric Equation: $x = a\cos\theta$, $y = b\sin\theta$ Eccentricity: $e < 1$ Focus: $(\pm ae, 0)$ Directrix: $x = \pm\frac{a}{e}$

Other Properties $c^2 = a^2e^2 = a^2 - b^2$ where $c$ is the focal length meaning the distance from the centre to a focus

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Flashcards

A-Level Further Pure 1

#z_Legacy/Maths/A-Levels/Further-Pure/Conics

How are is the general point $P$, the focus $F$, the directrix $D$ and eccentricity $e$ related? ? For all points $P$ on a conic section, the ratio of the distance $P$ from a fixed point $F$ (the focus) and fixed straight line $D$ (the directrix) is constant. This ratio $e$ is called the eccentricity.

• If $0 \leq e < 1$, $P$ describes an ellipse
• If $e = 1$, $P$ describes a parabola
• If $e > 1$, $P$ describes a hyperbola