8. Gira la cónica \[ 11 x^{2}-16 x y+15 y^{2}-1=0 \] de manera que ya no tenga término en \( x y \). Escribe los coeficientes redondeando a dos decimales.

Para eliminar el término \(xy\) de la ecuación de la cónica \(11 x^{2}-16 x y+15 y^{2}-1=0\), podemos utilizar una rotación de ejes. La fórmula para la rotación de ejes es: \[ x = x' \cos(\theta) - y' \sin(\theta) \] \[ y = x' \sin(\theta) + y' \cos(\theta) \] Donde \(\theta\) es el ángulo de rotación que se puede calcular usando la siguiente fórmula: \[ \tan(2\theta) = \frac{B}{A - C} \] En nuestra ecuación, los coeficientes son: - \(A = 11\) - \(B = -16\) - \(C = 15\) Primero, calculamos \(A - C\): \[ A - C = 11 - 15 = -4 \] Ahora, calculamos \(\tan(2\theta)\): \[ \tan(2\theta) = \frac{-16}{-4} = 4 \] Ahora, encontramos el ángulo \(2\theta\): \[ 2\theta = \tan^{-1}(4) \] Calculamos \(\theta\): \[ \theta = \frac{1}{2} \tan^{-1}(4) \] Ahora, vamos a calcular el valor de \(\theta\) y luego usaremos este ángulo para rotar la cónica. Primero, calculamos \(\tan^{-1}(4)\) y luego dividimos por 2. Voy a calcular \(\tan^{-1}(4)\) y luego \(\theta\). Calculate the value by following steps: - step0: Calculate: \(\arctan\left(4\right)\) Calculate or simplify the expression \( 1.325818/2 \). Calculate the value by following steps: - step0: Calculate: \(\frac{1.325818}{2}\) - step1: Convert the expressions: \(\frac{\frac{662909}{500000}}{2}\) - step2: Multiply by the reciprocal: \(\frac{662909}{500000}\times \frac{1}{2}\) - step3: Multiply the fractions: \(\frac{662909}{500000\times 2}\) - step4: Multiply: \(\frac{662909}{1000000}\) Hemos encontrado que: \[ \tan^{-1}(4) \approx 1.325818 \quad \text{y} \quad \theta \approx 0.662909 \] Ahora, utilizamos este ángulo para calcular los nuevos coeficientes de la cónica. Los nuevos coeficientes \(A'\), \(B'\) y \(C'\) después de la rotación se pueden calcular usando las siguientes fórmulas: \[ A' = A \cos^2(\theta) + B \cos(\theta) \sin(\theta) + C \sin^2(\theta) \] \[ B' = 0 \] \[ C' = A \sin^2(\theta) - B \cos(\theta) \sin(\theta) + C \cos^2(\theta) \] Primero, necesitamos calcular \(\cos(\theta)\) y \(\sin(\theta)\): \[ \cos(\theta) = \cos(0.662909) \] \[ \sin(\theta) = \sin(0.662909) \] Voy a calcular estos valores. Calculate the value by following steps: - step0: Calculate: \(\sin\left(0.662909\right)\) Calculate or simplify the expression \( \cos(0.662909) \). Calculate the value by following steps: - step0: Calculate: \(\cos\left(0.662909\right)\) Calculate or simplify the expression \( 11*(0.615412^2) - (-16)*(0.788205)*(0.615412) + 15*(0.788205^2) \). Calculate the value by following steps: - step0: Calculate: \(11\times 0.615412^{2}-\left(-16\right)\times 0.788205\times 0.615412+15\times 0.788205^{2}\) - step1: Remove the parentheses: \(11\times 0.615412^{2}-\left(-16\times 0.788205\times 0.615412\right)+15\times 0.788205^{2}\) - step2: Convert the expressions: \(11\left(\frac{153853}{250000}\right)^{2}-\left(-16\times 0.788205\times 0.615412\right)+15\times 0.788205^{2}\) - step3: Convert the expressions: \(11\left(\frac{153853}{250000}\right)^{2}-\left(-16\times 0.788205\times 0.615412\right)+15\left(\frac{157641}{200000}\right)^{2}\) - step4: Simplify: \(11\times \frac{153853^{2}}{250000^{2}}-\left(-16\times 0.788205\times 0.615412\right)+15\left(\frac{157641}{200000}\right)^{2}\) - step5: Multiply the terms: \(11\times \frac{153853^{2}}{250000^{2}}-\left(-\frac{24253540773}{3125000000}\right)+15\left(\frac{157641}{200000}\right)^{2}\) - step6: Simplify: \(11\times \frac{153853^{2}}{250000^{2}}-\left(-\frac{24253540773}{3125000000}\right)+15\times \frac{157641^{2}}{200000^{2}}\) - step7: Rewrite the expression: \(\frac{11\times 153853^{2}}{250000^{2}}-\left(-\frac{24253540773}{3125000000}\right)+15\times \frac{157641^{2}}{200000^{2}}\) - step8: Rewrite the expression: \(\frac{11\times 153853^{2}}{250000^{2}}-\left(-\frac{24253540773}{3125000000}\right)+\frac{15\times 157641^{2}}{200000^{2}}\) - step9: Rewrite the expression: \(\frac{11\times 153853^{2}}{250000^{2}}-\frac{-24253540773}{3125000000}+\frac{15\times 157641^{2}}{200000^{2}}\) - step10: Reduce fractions to a common denominator: \(\frac{11\times 153853^{2}\times 3125000000}{250000^{2}\times 3125000000}-\frac{-24253540773\times 250000^{2}}{3125000000\times 250000^{2}}+\frac{15\times 157641^{2}\times 4882812500}{200000^{2}\times 4882812500}\) - step11: Reorder the terms: \(\frac{11\times 153853^{2}\times 3125000000}{3125000000\times 250000^{2}}-\frac{-24253540773\times 250000^{2}}{3125000000\times 250000^{2}}+\frac{15\times 157641^{2}\times 4882812500}{200000^{2}\times 4882812500}\) - step12: Reorder the terms: \(\frac{11\times 153853^{2}\times 3125000000}{3125000000\times 250000^{2}}-\frac{-24253540773\times 250000^{2}}{3125000000\times 250000^{2}}+\frac{15\times 157641^{2}\times 4882812500}{4882812500\times 200000^{2}}\) - step13: Rewrite the expression: \(\frac{11\times 153853^{2}\times 3125000000}{3125000000\times 250000^{2}}-\frac{-24253540773\times 250000^{2}}{3125000000\times 250000^{2}}+\frac{15\times 157641^{2}\times 4882812500}{3125000000\times 250000^{2}}\) - step14: Transform the expression: \(\frac{11\times 153853^{2}\times 3125000000-\left(-24253540773\times 250000^{2}\right)+15\times 157641^{2}\times 4882812500}{3125000000\times 250000^{2}}\) - step15: Multiply the terms: \(\frac{34375000000\times 153853^{2}-\left(-24253540773\times 250000^{2}\right)+15\times 157641^{2}\times 4882812500}{3125000000\times 250000^{2}}\) - step16: Multiply the terms: \(\frac{34375000000\times 153853^{2}-\left(-24253540773\times 250000^{2}\right)+73242187500\times 157641^{2}}{3125000000\times 250000^{2}}\) - step17: Calculate: \(\frac{34375000000\times 153853^{2}+3416027975547\times 31250^{2}}{3125000000\times 250000^{2}}\) - step18: Rewrite the expression: \(\frac{195312500\left(176\times 153853^{2}+17080139877735\right)}{3125000000\times 250000^{2}}\) - step19: Reduce the fraction: \(\frac{176\times 153853^{2}+17080139877735}{16\times 250000^{2}}\) - step20: Calculate: \(\frac{176\times 153853^{2}+17080139877735}{1000^{4}}\) Calculate or simplify the expression \( 11*(0.788205^2) + (-16)*(0.788205)*(0.615412) + 15*(0.615412^2) \). Calculate the value by following steps: - step0: Calculate: \(11\times 0.788205^{2}+\left(-16\right)\times 0.788205\times 0.615412+15\times 0.615412^{2}\) - step1: Remove the parentheses: \(11\times 0.788205^{2}-16\times 0.788205\times 0.615412+15\times 0.615412^{2}\) - step2: Convert the expressions: \(11\left(\frac{157641}{200000}\right)^{2}-16\times 0.788205\times 0.615412+15\times 0.615412^{2}\) - step3: Convert the expressions: \(11\left(\frac{157641}{200000}\right)^{2}-16\times 0.788205\times 0.615412+15\left(\frac{153853}{250000}\right)^{2}\) - step4: Simplify: \(11\times \frac{157641^{2}}{200000^{2}}-16\times 0.788205\times 0.615412+15\left(\frac{153853}{250000}\right)^{2}\) - step5: Multiply the terms: \(11\times \frac{157641^{2}}{200000^{2}}-\frac{24253540773}{3125000000}+15\left(\frac{153853}{250000}\right)^{2}\) - step6: Simplify: \(11\times \frac{157641^{2}}{200000^{2}}-\frac{24253540773}{3125000000}+15\times \frac{153853^{2}}{250000^{2}}\) - step7: Rewrite the expression: \(\frac{11\times 157641^{2}}{200000^{2}}-\frac{24253540773}{3125000000}+15\times \frac{153853^{2}}{250000^{2}}\) - step8: Rewrite the expression: \(\frac{11\times 157641^{2}}{200000^{2}}-\frac{24253540773}{3125000000}+\frac{15\times 153853^{2}}{250000^{2}}\) - step9: Reduce fractions to a common denominator: \(\frac{11\times 157641^{2}\times 3125000000}{200000^{2}\times 3125000000}-\frac{24253540773\times 200000^{2}}{3125000000\times 200000^{2}}+\frac{15\times 153853^{2}\times 2000000000}{250000^{2}\times 2000000000}\) - step10: Reorder the terms: \(\frac{11\times 157641^{2}\times 3125000000}{3125000000\times 200000^{2}}-\frac{24253540773\times 200000^{2}}{3125000000\times 200000^{2}}+\frac{15\times 153853^{2}\times 2000000000}{250000^{2}\times 2000000000}\) - step11: Reorder the terms: \(\frac{11\times 157641^{2}\times 3125000000}{3125000000\times 200000^{2}}-\frac{24253540773\times 200000^{2}}{3125000000\times 200000^{2}}+\frac{15\times 153853^{2}\times 2000000000}{2000000000\times 250000^{2}}\) - step12: Rewrite the expression: \(\frac{11\times 157641^{2}\times 3125000000}{3125000000\times 200000^{2}}-\frac{24253540773\times 200000^{2}}{3125000000\times 200000^{2}}+\frac{15\times 153853^{2}\times 2000000000}{3125000000\times 200000^{2}}\) - step13: Transform the expression: \(\frac{11\times 157641^{2}\times 3125000000-24253540773\times 200000^{2}+15\times 153853^{2}\times 2000000000}{3125000000\times 200000^{2}}\) - step14: Multiply the terms: \(\frac{34375000000\times 157641^{2}-24253540773\times 200000^{2}+15\times 153853^{2}\times 2000000000}{3125000000\times 200000^{2}}\) - step15: Multiply the terms: \(\frac{34375000000\times 157641^{2}-24253540773\times 200000^{2}+30000000000\times 153853^{2}}{3125000000\times 200000^{2}}\) - step16: Subtract the numbers: \(\frac{-185438941017\times 25000^{2}+30000000000\times 153853^{2}}{3125000000\times 200000^{2}}\) - step17: Rewrite the expression: \(\frac{25000^{2}\left(-185438941017+48\times 153853^{2}\right)}{3125000000\times 200000^{2}}\) - step18: Reduce the fraction: \(\frac{-185438941017+48\times 153853^{2}}{5\times 200000^{2}}\) Hemos calculado los valores de \(\cos(\theta)\) y \(\sin(\theta)\): \[ \cos(0.662909) \approx 0.788205 \] \[ \sin(0.662909) \approx 0.615412 \] Ahora, usando estos valores, calculamos los nuevos coeficientes \(A'\) y \(C'\): 1. **Cálculo de \(A'\)**: \[ A' = 11 \cdot (0.788205^2) + (-16) \cdot (0.788205) \cdot (0.615412) + 15 \cdot (0.615412^2) \approx 4.75 \] 2. **Cálculo de \(C'\)**: \[ C' = 11 \cdot (0.615412^2) - (-16) \cdot (0.788205) \cdot (0.615412) + 15 \cdot (0.788205^2) \approx 21.25 \] Por lo tanto, los nuevos coeficientes de la cónica, redondeando a dos decimales, son: - \(A' \approx 4.75\) - \(B' = 0\) - \(C' \approx 21.25\) La ecuación de la cónica después de la rotación es: \[ 4.75 x'^2 + 21.25 y'^2 - 1 = 0 \]