Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola.
-$$5 x ^ { 2 } + 7 x y + 5 y ^ { 2 } + 2 x - 7 y = 0$$
A) Ellipse
B) Parabola
C) Hyperbola

Question

Quizplus · Accepted Answer

The given equation is of the form $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, where $A = 5$, $B = 7$, $C = 5$, $D = 2$, $E = -7$, and $F$ is implicitly $0$. The discriminant for conic sections is given by $B^2 - 4AC$. Substituting the given values, we get $7^2 - 4(5)(5) = 49 - 100 = -51$. Since the discriminant is negative ($-51 < 0$), the equation represents an ellipse.

Use the Discriminant to Decide Whether the Equation Represents a Parabola