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.

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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.

La ecuación de la cónica rotada sin término $$ x y $$ es: $$ -14.00x'^{2} - 4.00y'^{2} + 11.00 = 0 $$ Los coeficientes redondeados a dos decimales son: - $$ A = -14.00 $$ - $$ B = 0.00 $$ - $$ C = -4.00 $$ - $$ D = 0.00 $$ - $$ E = 0.00 $$ - $$ F = 11.00 $$

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.

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