Determine the angle $\theta$ through which the axes are rotated.Round your answer to two decimal places. $48 x ^ { 2 } + 69 x y + 25 y ^ { 2 } = 50$
A) $\theta = 78.87 ^ { \circ }$
B) $\theta = 77.87 ^ { \circ }$
C) $\theta = 79.87 ^ { \circ }$
D) $\theta = 81.87 ^ { \circ }$
E) $\theta = 80.87 ^ { \circ }$

Question

Determine the angle $\theta$ through which the axes are rotated.Round your answer to two decimal places.​ $48 x ^ { 2 } + 69 x y + 25 y ^ { 2 } = 50$ ​
A) $\theta = 78.87 ^ { \circ }$
B) $\theta = 77.87 ^ { \circ }$
C) $\theta = 79.87 ^ { \circ }$
D) $\theta = 81.87 ^ { \circ }$
E) $\theta = 80.87 ^ { \circ }$

Quizplus · Accepted Answer

The angle of rotation $\theta$ for the axes to eliminate the $xy$ term in a quadratic equation of the form $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ can be found using the formula $\theta = \frac{1}{2} \tan^{-1}\left(\frac{B}{A-C}\right)$. For the given equation $48x^2 + 69xy + 25y^2 = 50$, we have $A = 48$, $B = 69$, and $C = 25$. Plugging these values into the formula gives $\theta = \frac{1}{2} \tan^{-1}\left(\frac{69}{48-25}\right) = \frac{1}{2} \tan^{-1}\left(\frac{69}{23}\right)$. Calculating this gives $\theta \approx 77.87^\circ$.

Determine the Angle θ\thetaθ Through Which the Axes Are Rotated 48x2+69xy+25y2=5048 x ^ { 2 } + 69 x y + 25 y ^ { 2 } = 5048x2+69xy+25y2=50

Determine the Angle $\theta$ Through Which the Axes Are Rotated $48 x ^ { 2 } + 69 x y + 25 y ^ { 2 } = 50$