CSPE - Hyperbolas Lesson

Hyperbolas

image of a plane slicing a cone to form a hyperbola

The last type of conic section we will explore is hyperbolas. A hyperbola is a curve with two branches. A hyperbola is defined as the locus of points P (x, y) such that the difference of the distance from P to two fixed points, called the foci, is constant. The key pieces of a hyperbola that we will identify are the center, vertices, and foci.

Standard Form Equations for Hyperbolas

$LaTeX: \frac{\left(x-h\right)^2}{a^2}-\frac{\left(y-k\right)^2}{b^2}=1$ $\frac{\left(x-h\right)^2}{a^2}-\frac{\left(y-k\right)^2}{b^2}=1$	$LaTeX: \frac{\left(y-k\right)^2}{a^2}-\frac{\left(x-h\right)^2}{b^2}=1$ $\frac{\left(y-k\right)^2}{a^2}-\frac{\left(x-h\right)^2}{b^2}=1$

Opens Left/Right (Horizontal Transverse Axis)	Opens Up/Down (Vertical Transverse Axis)
Center: (h, k)	Center: (h, k)
Vertices: $LaTeX: \left(h\pm a,\:k\right)$ $\left(h\pm a,\:k\right)$	Vertices: $LaTeX: \left(h,\:k\pm a\right)$ $\left(h,\:k\pm a\right)$
Foci: $LaTeX: \left(h\pm c,\:k\right)$ $\left(h\pm c,\:k\right)$	Foci: $LaTeX: \left(h,\:k\pm c\right)$ $\left(h,\:k\pm c\right)$
Asymptote: $LaTeX: \left(y-k\right)=\pm\frac{b}{a}\left(x-h\right)$ $\left(y-k\right)=\pm\frac{b}{a}\left(x-h\right)$	Asymptote: $LaTeX: \left(y-k\right)=\pm\frac{a}{b}\left(x-h\right)$ $\left(y-k\right)=\pm\frac{a}{b}\left(x-h\right)$
The equation for finding c: $c^2=a^2+b^2$

Let's try writing the equation given the graph of a hyperbola.

Watch the following videos to practice graphing hyperbolas and converting forms.

Given the location of the vertices and foci - give the equation of the hyperbolas. Hint: sketch the graph!

Vertices: (0, 10), (0, -10) Foci: equation image indicator $LaTeX: \left(0,\:2\sqrt[]{41}\right),\:\left(0,\:-2\sqrt[]{41}\right)$ $\left(0,\:2\sqrt[]{41}\right),\:\left(0,\:-2\sqrt[]{41}\right)$ text annotation indicator

Solution: $LaTeX: \frac{y^2}{100}-\frac{x^2}{64}=1$ $\frac{y^2}{100}-\frac{x^2}{64}=1$

Vertices: (3, 0), (-3, 0) Foci: $LaTeX: \left(\sqrt[]{34},\:0\right),\:\left(-\sqrt[]{34},\:0\right)$ $\left(\sqrt[]{34},\:0\right),\:\left(-\sqrt[]{34},\:0\right)$ text annotation indicator

Solution: $LaTeX: \frac{x^2}{9}-\frac{y^2}{25}=1$ $\frac{x^2}{9}-\frac{y^2}{25}=1$

Vertices: (0, 8), (0, -8) Foci: equation image indicator $LaTeX: \left(0,\:\sqrt[]{185}\right),\:\left(0,\:-\sqrt[]{185}\right)$ $\left(0,\:\sqrt[]{185}\right),\:\left(0,\:-\sqrt[]{185}\right)$ text annotation indicator

Solution: $LaTeX: \frac{y^2}{64}-\frac{x^2}{121}=1$ $\frac{y^2}{64}-\frac{x^2}{121}=1$

Vertices: equation image indicator $LaTeX: \left(\sqrt[]{55},\:0\right),\:\left(-\sqrt[]{55},\:0\right)$ $\left(\sqrt[]{55},\:0\right),\:\left(-\sqrt[]{55},\:0\right)$ Foci: $LaTeX: \left(\sqrt[]{110},\:0\right),\:\left(-\sqrt[]{110},\:0\right)$ $\left(\sqrt[]{110},\:0\right),\:\left(-\sqrt[]{110},\:0\right)$ text annotation indicator

Solution: $LaTeX: \frac{x^2}{55}-\frac{y^2}{55}=1$ $\frac{x^2}{55}-\frac{y^2}{55}=1$

Properties of Hyperbolas Practice

Given the equation of each hyperbola, find the correct properties. Then sketch the graph on your own paper with the requested information.

Now that you've investigated the general forms of conic sections, let's think about how we might identify the type of conic by just looking at the general equation.

equation image indicator LaTeX: Ax^2+Cy^2+Dx+Ey+F=0 $Ax^2+Cy^2+Dx+Ey+F=0$

You know it's going to be a parabola if only x or y is squared, A = 0, or C = 0
You know it's going to be a hyperbola if either A is negative or C is negative, but not both and neither can be 0.
You know it's going to be a circle if $LaTeX: A=C\ne0$ $A=C\ne0$ .
You know it's going to be an ellipse if $LaTeX: A\ne C$ $A\ne C$ , but they are the same sign and neither can be 0.

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