12.2 The hyperbola  (Page 8/13)

A hyperbola is the set of points in a plane the difference of whose distances from two fixed points (foci) is a positive constant.

The foci must lie on the transverse axis and be in the interior of the hyperbola.

The center must be the midpoint of the line segment joining the foci.

yes $\frac{x^2}{6^2}-\frac{y^2}{3^2}=1$

yes $\frac{x^2}{4^2}-\frac{y^2}{5^2}=1$

$\frac{x^2}{5^2}-\frac{y^2}{6^2}=1;$ vertices: $\left(5,0\right),\left(-5,0\right);$ foci: $\left(\sqrt{61},0\right),\left(-\sqrt{61},0\right);$ asymptotes: $y=\frac{6}{5}x,y=-\frac{6}{5}x$

$\frac{y^2}{2^2}-\frac{x^2}{9^2}=1;$ vertices: $\left(0,2\right),\left(0,-2\right);$ foci: $\left(0,\sqrt{85}\right),\left(0,-\sqrt{85}\right);$ asymptotes: $y=\frac{2}{9}x,y=-\frac{2}{9}x$

$\frac{{\left(x-1\right)}^2}{3^2}-\frac{{\left(y-2\right)}^2}{4^2}=1;$ vertices: $\left(4,2\right),\left(-2,2\right);$ foci: $\left(6,2\right),\left(-4,2\right);$ asymptotes: $y=\frac{4}{3}\left(x-1\right)+2,y=-\frac{4}{3}\left(x-1\right)+2$