Identify the equation without applying a rotation of axes.
-$$x ^ { 2 } - 3 x y - 3 y ^ { 2 } - 2 x - 3 y - 5 = 0$$
A) parabola
B) ellipse
C) hyperbola
D) not a conic

Question

Quizplus · Accepted Answer

The given equation is a hyperbola. This can be determined without rotating axes by examining the coefficients of $x^2$, $xy$, and $y^2$. The presence of the $xy$ term suggests an orientation that is not aligned with the coordinate axes, but the key is the signs and relative magnitudes of the $x^2$ and $y^2$ terms. Since the coefficients of $x^2$ and $y^2$ are of opposite signs (1 and -3, respectively), this indicates a hyperbola. The equation does not represent a parabola, ellipse, or a non-conic curve based on these coefficients.

Identify the Equation Without Applying a Rotation of Axes x2−3xy−3y2−2x−3y−5=0x ^ { 2 } - 3 x y - 3 y ^ { 2 } - 2 x - 3 y - 5 = 0x2−3xy−3y2−2x−3y−5=0

Identify the Equation Without Applying a Rotation of Axes $x ^ { 2 } - 3 x y - 3 y ^ { 2 } - 2 x - 3 y - 5 = 0$