12.2 The hyperbola  (Page 3/13)

Given the equation of a hyperbola in standard form, locate its vertices and foci.

1. Determine whether the transverse axis lies on the x - or y -axis. Notice that $a^2$ is always under the variable with the positive coefficient. So, if you set the other variable equal to zero, you can easily find the intercepts. In the case where the hyperbola is centered at the origin, the intercepts coincide with the vertices.
   1. If the equation has the form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,$ then the transverse axis lies on the x -axis. The vertices are located at $(\pm a,0),$ and the foci are located at $\left(\pm c,0\right).$
   2. If the equation has the form $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1,$ then the transverse axis lies on the y -axis. The vertices are located at $(0,\pm a),$ and the foci are located at $\left(\mathrm{0,}\pm c\right).$
2. Solve for $a$ using the equation $a=\sqrt{a^2}.$
3. Solve for $c$ using the equation $c=\sqrt{a^2+b^2}.$

Given the vertices and foci of a hyperbola centered at $\left(0,\text{0}\right),$ write its equation in standard form.

1. Determine whether the transverse axis lies on the x - or y -axis.
   1. If the given coordinates of the vertices and foci have the form $\left(\pm a,0\right)$ and $\left(\pm c,0\right),$ respectively, then the transverse axis is the x -axis. Use the standard form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$
   2. If the given coordinates of the vertices and foci have the form $\left(\mathrm{0,}\pm a\right)$ and $\left(\mathrm{0,}\pm c\right),$ respectively, then the transverse axis is the y -axis. Use the standard form $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1.$
2. Find $b^2$ using the equation $b^2=c^2-a^2.$
3. Substitute the values for $a^2$ and $b^2$ into the standard form of the equation determined in Step 1.