Identify the equation without applying a rotation of axes.
-$$2 x ^ { 2 } + 8 x y + 16 y ^ { 2 } - 3 x - 2 y + 10 = 0$$
A) parabola
B) hyperbola
C) ellipse
D) not a conic

Question

Quizplus · Accepted Answer

The given equation is a conic section, and to determine its type without rotating axes, we can use the discriminant method for conic sections: $B^2 - 4AC$, where $A$, $B$, and $C$ are coefficients from the general conic equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$. Here, $A = 2$, $B = 8$, and $C = 16$, so $B^2 - 4AC = 8^2 - 4(2)(16) = 64 - 128 = -64$. Since the discriminant is negative, the conic is an ellipse.

Identify the Equation Without Applying a Rotation of Axes 2x2+8xy+16y2−3x−2y+10=02 x ^ { 2 } + 8 x y + 16 y ^ { 2 } - 3 x - 2 y + 10 = 02x2+8xy+16y2−3x−2y+10=0

Identify the Equation Without Applying a Rotation of Axes $2 x ^ { 2 } + 8 x y + 16 y ^ { 2 } - 3 x - 2 y + 10 = 0$