Gira la cónica \[ -14 x^{2}-x y-4 y^{2}+11=0 \] e manera que ya no tenga término en \( x y \). Escribe s coeficientes redondeando a dos decimales.

Okay, I need to rotate the conic section given by the equation: \[ -14x^{2} - xy - 4y^{2} + 11 = 0 \] so that it no longer has a \( xy \) term. Then, I need to write down the coefficients, rounding them to two decimal places. First, I recall that to eliminate the \( xy \) term in a general second-degree equation of the form: \[ Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0 \] we can use a rotation of axes. The angle of rotation \( \theta \) that eliminates the \( xy \) term is given by: \[ \tan(2\theta) = \frac{B}{A - C} \] In this case, \( A = -14 \), \( B = -1 \), and \( C = -4 \). Plugging these values into the formula: \[ \tan(2\theta) = \frac{-1}{-14 - (-4)} = \frac{-1}{-10} = 0.1 \] So, \( 2\theta = \arctan(0.1) \). Calculating \( \arctan(0.1) \) gives approximately \( 5.71^\circ \). Therefore, \( \theta \approx 2.855^\circ \). Next, I need to find the rotation matrix. The rotation matrix \( R \) for an angle \( \theta \) is: \[ R = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} \] Using \( \theta \approx 2.855^\circ \), I can calculate \( \cos\theta \) and \( \sin\theta \). However, since the angle is small, I can approximate \( \cos\theta \approx 1 \) and \( \sin\theta \approx \theta \) in radians. Converting \( 2.855^\circ \) to radians: \[ \theta \approx 2.855 \times \frac{\pi}{180} \approx 0.0497 \text{ radians} \] So, \( \cos\theta \approx 1 \) and \( \sin\theta \approx 0.0497 \). Now, applying the rotation to the original equation. The new coordinates \( x' \) and \( y' \) are related to the original coordinates \( x \) and \( y \) by: \[ \begin{cases} x = x'\cos\theta - y'\sin\theta \\ y = x'\sin\theta + y'\cos\theta \end{cases} \] Substituting these into the original equation and simplifying will eliminate the \( x'y' \) term. After performing these substitutions and simplifications, the equation should no longer contain the \( x'y' \) term. Finally, I need to write down the coefficients of the new equation, rounding them to two decimal places. This involves identifying the coefficients of \( x'^2 \), \( y'^2 \), \( x' \), \( y' \), and the constant term in the rotated equation.