Question

How do you identify the following equation `xy = 4` as that of a line, a circle, an ellipse, a parabola, or a hyperbola.?

Answer 1

It is a hyperbola.

Explanation:

The general second degree equation can be written as
`Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0`

To identify the graph by inspection of equation complete any squares if necessary so that the variables are on one side and the constant is on the right hand side of the equation. Check against the following:

1. Line
   Squared term of neither variable is present.
2. Circle
   Squared terms of both variables are present, both are positive and both have the same coefficient. The right hand side is positive.
3. Ellipse
   Squared terms of both variables are present, both are positive and both have different coefficients. The right hand side is positive.
4. Parabola
   Both variables are present but only one is squared.
5. Hyperbola
   Squared terms of both variables are present, but one is positive and the other is negative. The coefficients may or may not be the same. The right hand side is not zero.

For 3, 4, and 5.
The general equation reduces to `Ax^2 + Bxy + Cy^2 = 1`
which is a hyperbola if `B^2 > 4AC` and an ellipse if `B^2 < 4AC`.
A parabola has `B^2 = 4AC`.
For the given equation `xy=4`, we observe that `B^2 > 4AC`. Hence it is a hyperbola.

graph{xy=4 [-10, 10, -5, 5]}