Determine the angle $\theta$ through which the axes are rotated.Round your answer to two decimal places. $x ^ { 2 } + 2 x y + y ^ { 2 } = 10$
A) $\theta = 45 ^ { \circ }$
B) $\theta = 60 ^ { \circ }$
C) $\theta = 90 ^ { \circ }$
D) $\theta = 145 ^ { \circ }$
E) $\theta = 30 ^ { \circ }$

Question

Determine the angle $\theta$ through which the axes are rotated.Round your answer to two decimal places.  $x ^ { 2 } + 2 x y + y ^ { 2 } = 10$ 
A) $\theta = 45 ^ { \circ }$
B) $\theta = 60 ^ { \circ }$
C) $\theta = 90 ^ { \circ }$
D) $\theta = 145 ^ { \circ }$
E) $\theta = 30 ^ { \circ }$

Quizplus · Accepted Answer

The given equation is in the form of a rotated conic section. The coefficient of $xy$ (which is 2 in this case) determines the angle of rotation $\theta$ using the formula $\tan(2\theta) = \frac{2B}{A-C}$, where $A$, $B$, and $C$ are coefficients from the general conic section equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$. Here, $A = 1$, $B = 1$, and thus $\tan(2\theta) = 2$, which gives $2\theta = 90^\circ$, or $\theta = 45^\circ$.

Determine the Angle θ\thetaθ Through Which the Axes Are Rotated x2+2xy+y2=10x ^ { 2 } + 2 x y + y ^ { 2 } = 10x2+2xy+y2=10

Determine the Angle $\theta$ Through Which the Axes Are Rotated $x ^ { 2 } + 2 x y + y ^ { 2 } = 10$