Final answer:
To remove the xy term from the equation 21x² + 10√3xy − 31y² − 144 = 0 using a rotation of axes, one must find the angle of rotation that zeroes the coefficient of the XY term in the transformed equation.
Step-by-step explanation:
The question is asking to eliminate the xy term from the quadratic equation by using a rotation of axes. The equation given is 21x² + 10√3xy − 31y² − 144 = 0. The rotational transformation to new coordinates (X, Y) can be represented by x = Xcos(θ) + Ysin(θ) and y = −Xsin(θ) + Ycos(θ), where θ is the angle of rotation.
To eliminate the xy term, we need to find the angle θ for which the coefficient of the XY term in the transformed equation becomes zero. Using the formula cot(2θ) = (A - C) / B, where A, B, and C are the coefficients of x², xy, and y² in the original equation, respectively. In this case, the formula becomes cot(2θ) = (21 + 31) / (10√3).
Once θ is found, using this rotation, the equation becomes an expression in X and Y without the XY term, allowing us to classify the conic section and solve the system more easily.