audio read-through Equations of Hyperbolas in Standard Form (Part 2)

(This page is Part 2. Click here for Part 1.)

Be Careful!

The variables $\,a\,,$ $\,b\,$ and $\,c\,$ are used for both ellipses and hyperbolas, but their relationship is different!

For ellipses, $\,c^2 = a^2 \color{red}{-} b^2\,.$

For hyperbolas, $\,c^2 = a^2 \color{red}{+} b^2\,.$

Derivation of the Equation of a Hyperbola: Center at the Origin, Foci on the $y$-axis

The derivation of the equation of a hyperbola with center at the origin and foci on the $y$-axis is nearly identical to the derivation in Part 1.


... the Derivation of the Hyperbola With Center at the Origin and Foci on the Vertical Axis

Summary:  Equations of Hyperbolas with Center at the Origin and Foci on the $x$-axis or $y$-axis

In Both Cases:

$0 \lt a \lt c$

With the equations in standard form, the number $\,a^2\,$ is the denominator of the positive term.

The hyperbola constant is $\,2a\,.$

The foci are determined by solving the equation $\,c^2 = a^2 + b^2\,$ for $\,c\,.$

Be Careful!

The variables $\,a\,,$ $\,b\,$ and $\,c\,$ are used for both ellipses and hyperbolas, but their relationship is different!

For ellipses, $\,c^2 = a^2 \color{red}{-} b^2\,.$

For hyperbolas, $\,c^2 = a^2 \color{red}{+} b^2\,.$

Foci on the $x$-axis

hyperbola with foci on x-axis

Equation of Hyperbola:

$$\cssId{s23}{\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1}$$

When the foci are on the $\color{red}{x}$-axis, the $\,\color{red}{x^2}$-term is positive.

Coordinates of foci: $\,(-c,0)\,$ and $\,(c,0)$

Foci on the $y$-axis

hyperbola with foci on y-axis

Equation of Hyperbola:

$$\cssId{s30}{\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1}$$

When the foci are on the $\color{red}{y}$-axis, the $\,\color{red}{y^2}$-term is positive.

Coordinates of foci: $\,(0,-c)\,$ and $\,(0,c)$

Tips

The key to recognizing the equation of a hyperbola with center at the origin and foci on either the $x$-axis or $y$-axis is this:

In both equations, the vertices are easy to find. Since the vertices and foci are both on the same axis (the major axis), this is a good way to minimize memorization and let the equation ‘tell you’ where the foci lie:

To find the foci: With the equations in standard form, $\,c^2\,$ is always the sum of the two denominators:

$$ \begin{align} c^2 &= (\text{denominator of $\,x^2\,$ term})\cr &\quad + (\text{denominator of $\,y^2\,$ term}) \end{align} $$

Example: Finding the Equation of a Hyperbola

Find the Equation of the Following Hyperbola:

Also, find the coordinates of the foci.

Solution

Since the major axis is along the $y$-axis, the form of the equation is:

$$ \cssId{s59}{\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1} $$

$\,a = 5\,$

Since $\,(3,5\sqrt{2})\,$ is on the graph:

$$ \begin{gather} \cssId{s62}{\frac{(5\sqrt{2})^2}{5^2} - \frac{3^2}{b^2} = 1}\cr\cr \cssId{s63}{2 - \frac{9}{b^2} = 1}\cr\cr \cssId{s64}{\frac{9}{b^2} = 1}\cr\cr \cssId{s65}{b^2 = 9} \end{gather} $$

The equation is:

$$ \cssId{s67}{\frac{y^2}{25} - \frac{x^2}{9} = 1} $$

Foci:

$$ \begin{align} c^2 &= a^2 + b^2\cr &= 25 + 9\cr &= 34 \end{align} $$

so $\,c = \sqrt{34}\,.$

Coordinates of foci:

$$ (0,-\sqrt{34})\ \ \text{and}\ \ (0,\sqrt{34}) $$

(Note that $\,5 \lt \sqrt{34}\,,$ so the foci are ‘inside’ the hyperbola.)

Concept Practice