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 represents an ellipse. This can be determined without rotating the axes by examining the coefficients of $x^2$, $xy$, and $y^2$. The equation is of the form $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, where $A = 2$, $B = 8$, and $C = 16$. For an equation to represent an ellipse (without considering a rotation), the condition $B^2 - 4AC < 0$ must be satisfied. Here, $8^2 - 4(2)(16) = 64 - 128 = -64 < 0$, which confirms the shape 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 $2 x ^ { 2 } + 8 x y + 16 y ^ { 2 } - 3 x - 2 y + 10 = 0$