Question

How do you identify the following equation `x^2 - y^2 = 4` as that of a line, a circle, an ellipse, a parabola, or a hyperbola.?

Answer 1

Rewrite the equation in General Cartesian Form and then use the conditions of the discriminant to make the identification.

Explanation:

From the reference, the General Cartesian Form is:

`Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0`

The Discriminant is:

`B^2 - 4AC`

The conditions stated in the reference are:

If `B^2 - 4AC < 0`, then the equation represents an ellipse.
A special case of this is, if `A = C = 1 and B = 0`, then the equation represents a circle.

If `B^2 - 4AC = 0`, then the equation represents a parabola.

If `B^2 - 4AC > 0`, then the equation represents a hyperbola.
A special case of this is, if A + C = 0, then the equation represents a rectangular hyperbola.

Rewrite the given equation in General Cartesian Form:

`x^2 - y^2 - 4 = 0`

Matching the coefficients in the general form:

`A = 1, C = -1, F = -4, and B = D = E = 0`

Compute the discriminant:

`B^2 - 4AC = 0^2 - 4(1)(-1) = 4`

The discriminant is greater than 0, therefore, the equation represents a hyperbola but, because `A + C = 1 - 1 = 0`, then we determine that the equation represents a Rectangular Hyperbola