Example 3. Show that the graph of equation is a hyperbola. Find the coordinates of the centre and vertices, equations of asymptotes, and sketch the graph.
Solution. The given equation can be rewritten in the form
Completing the squares in the last equation we obtain
or
and dividing the last equation by , we have
and this is the standard form of the equation for a hyperbola with the centre , horizontal major axis and semiaxes .
The distance between the centre and the foci is given by
thus the foci of the hyperbola have the coordinates
and the vertices are at points
Equations of asymptotes are
The graph is sketched in figure.
Figure 1: Graph of hyperbola |