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From our explorations, we know that if A and B have different signs, the equation Ax2 + By2 = P defines a hyperbola centred at the origin.

It is very difficult to determine the characteristics of the graph of a hyperbola given only this form of the equation. Therefore, if we write Ax2 + By2 = P in standard form, we could then use the equation to describe the characteristics of the graph.

There are two different instances when the equation Ax2 + By2 = P represents a hyperbola. The first case is when A > 0 and B < 0. The second case is when A < 0 and B > 0.

Let's first look at the case when A > 0, B < 0 and P > 0. Click on the "Example 1" button below and follow through the questions in the pop-up window to arrive at the standard form.

In the example you just looked at, the equation 9x2 - 4y2 = 36 was rewritten into an equation of the form x2/a2 - y2/b2 = 1. The latter form gives more useful information about the graph of the hyperbola.

This form of the equation is called the standard form of an equation for a hyperbola centred at the origin that opens left and right.

Now, click on the "Example 2" button to look at an example of a hyperbola of the form Ax2 + By2 = P, but with A < 0, B > 0 and P > 0. Hyperbolas of this form open up and down.

In Example 2, the equation -4x2 + 3y2 = 12 was rewritten into an equation of the form -x2/ a2 + y2/ b2 = 1.

As before, this form of the equation is called the standard form for a hyperbola centred at the origin that opens up and down, and this form gives us much useful information about the graph of the hyperbola.

Now, look at a summary of the standard equations of hyperbolas, and how to find the locations of the vertices.