1DMoreHyperbolas

// Pronounced hi-PER-bo-lah (emphasise the second syllable) //
 * Hyperbolas with Oblique Asymptotes**

Hyperbolas are one of the group of shapes called Conic Sections (not in course). Hyperbolas can be defined geometrically (not in course). A hyperbola has practical applications in Physics (not in course).

You are familiar with hyperbolas that have vertical and horizontal asymptotes. In this section we examine hyperbolas with __**oblique**__(sloping) asymptotes.

A hyperbola with asymptotes that cross at the origin can be defined using coordinate geometry as the relation: math . \qquad \dfrac {x^2}{a^2} - \dfrac{y^2}{b^2} = 1 math

(a and b are lengths so must be positive) The equations of the two asymptotes will be: math . \qquad y = \pm \dfrac {b}{a} x math

The hyperbola will have vertices at (–a, 0) and (a, 0)

math \text {Domain: } \{ x \leqslant -a \} \cup \{ x \geqslant a \} math

math \text {Range: } y \in R math

x-intercepts x = –a, x = a

y-intercepts none

Notice that the hyperbola is symmetrical in two directions:
 * across the x-axis
 * across the y-axis.

General Equation for an Hyperbola

If the hyperbola is translated h units to the right and k units up, we get the general equation for an hyperbola: math . \qquad \dfrac {(x-h)^2}{a^2} - \dfrac{(y-k)^2}{b^2} = 1 math

Asymptotes: math . \qquad y = \pm \dfrac {b}{a} (x - h) + k math

Asymptotes cross at (h, k) Vertices (h – a, k) and (h + a, k)

math \text {Domain: } \{ x \leqslant h-a \} \cup \{ x \geqslant h+a \} math

math \text {Range: } x \in R math

Example 1 math . \qquad \dfrac {(x-2)^2}{4} - \dfrac{(y-1)^2}{9} = 1 math This is an hyperbola centered at (2, 1) with a ``=`` 2, b ``=`` 3

Asymptotes: math . \qquad y = \pm \dfrac {3}{2} (x - 2) + 1 \\ math Or math \\ . \qquad y = \dfrac {3}{2}x - 2 \: \text {and} \\ . \qquad y = -\dfrac {3}{2}x + 4 math

Vertices (0, 1) and (4, 1)

math \text {Domain: } \lbrace x \leqslant 0 \rbrace \cup \lbrace x \geqslant 4 \rbrace math

math \text {Range: } y \in R math

x-intercepts: math . \qquad x = 2 \pm \dfrac {2\sqrt{10}}{3} math

y-intercepts: math . \qquad y = 1 math

Sketching hyperbolas on a Calculator

The best way to draw hyperbolas is to use parametric mode and the parametric equations of an hyperbola.

Return to summary of Conic Sections

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