# 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.?

Dec 30, 2016

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:

$A {x}^{2} + B x y + C {y}^{2} + D x + E y + F = 0$

The Discriminant is:

${B}^{2} - 4 A C$

The conditions stated in the reference are:

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

If ${B}^{2} - 4 A C = 0$, then the equation represents a parabola.

If ${B}^{2} - 4 A C > 0$, 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:

$A = 1 , C = - 1 , F = - 4 , \mathmr{and} B = D = E = 0$

Compute the discriminant:

${B}^{2} - 4 A C = {0}^{2} - 4 \left(1\right) \left(- 1\right) = 4$

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