# How do use the discriminant test to determine whether the graph x^2+5xy+1-y^2-16=0 whether the graph is parabola, ellipse, or hyperbola?

Jul 20, 2018

It is a hyperbola.

#### Explanation:

Let the equation be of the type $A {x}^{2} + B x y + C {y}^{2} + D x + E y + F = 0$

then if

${B}^{2} - 4 A C = 0$ and $A = 0$ or $C = 0$, it is a parabola

${B}^{2} - 4 A C < 0$ and $A = C$, it is a circle

${B}^{2} - 4 A C < 0$ and $A \ne C$, it is an ellipse

${B}^{2} - 4 A C > 0$, it is a hyperbola

In the given equation ${x}^{2} + 5 x y + 1 - {y}^{2} - 16 = 0$ or ${x}^{2} + 5 x y - {y}^{2} - 15 = 0$

$A = 1$, $B = 5$ and $C = - 1$

Therefore, ${B}^{2} - 4 A C = 25 + 4 = 29$

Hence, it is a hyperbola.

graph{x^2+5xy+1-y^2-16=0 [-20, 20, -10, 10]}