Step By Step Calculus » 3.4 - Hyperbola

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A curve, traced out by a point that moves in a plane so that the difference of its distances from two fixed points is always a constant, is called a hyperbola. These two fixed points are called the foci of hyperbola. A hyperbola with its two foci at (\pm p,0) (\pm p,0) is expressed by the equation

\displaystyle {\textbf{(Equation of Hyperbola)}\qquad\qquad \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \;\;\textrm{ where }p^2=a^2+b^2.}\displaystyle {\textbf{(Equation of Hyperbola)}\qquad\qquad \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \;\;\textrm{ where }p^2=a^2+b^2.}

The standard form of a general hyperbola is:

\displaystyle {\textbf{(Standard Form)}\;{\left\{\begin{array}{ll} \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1 & \textrm{ when foci on a horizontal line}\\ \frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1 & \textrm{ when foci on a vertical line}\end{array}\right.,}}\displaystyle {\textbf{(Standard Form)}\;{\left\{\begin{array}{ll} \frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1 & \textrm{ when foci on a horizontal line}\\ \frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1 & \textrm{ when foci on a vertical line}\end{array}\right.,}}

where (h,k)(h,k) is the center of the hyperbola and p^2=a^2+b^2p^2=a^2+b^2. In the first case, the foci are at \left(h-p,k\right)\left(h-p,k\right) and \left(h+p,k\right)\left(h+p,k\right) while they are at \left(h,k-p\right)\left(h,k-p\right) and \left(h,k+p\right)\left(h,k+p\right) in the second case.
A hyperbola has two branches, with a vertex on each branch and an axis that goes between the vertices. This axis is called the Transverse axis. If extended, the transverse axis would pass through the foci. It has length 2a2a. The line segment perpendicular to and centered on the transverse axis, with length 2b2b, is called the Conjugate axis.
The asymptotes to a hyperbola are the straight lines given by the formulas
\displaystyle \displaystyle

\displaystyle y=k\pm\frac{b}{a}(x-h) \quad\displaystyle y=k\pm\frac{b}{a}(x-h) \quadfor hyperbola with horizontal transverse axis,
\displaystyle \displaystyle

\displaystyle y=k\pm\frac{a}{b}(x-h) \quad\displaystyle y=k\pm\frac{a}{b}(x-h) \quadfor hyperbola with vertical transverse axis.\displaystyle \displaystyle
We use the following STAPLED procedure when sketching a hyperbola.
Standard Convert the given equation to standard form.
Type Indicate whether vertical or horizontal transverse axis.
Axes Draw a pair of xy xy-axes on the xy xy-plane. Use the same scale for both.
Points Mark the center and foci of the hyperbola on the xy xy-plane.
Lines Draw the Transverse Axes line segment and the Asymptotes, making sure they all pass through the center of the hyperbola.
Ends Make the ends of the Transverse axes clear, i.e. mark the vertices.
Draw Draw the hyperbola: Make sure it goes through the ends of the Transverse Axis. Also, ensure that the difference of the distances from each point on the hyperbola to the two foci is constant.