Question

Find in degrees the numeric value of the acute angle of rotation that eliminates the product term from the equation 7x2+24xy−34x+24y−185=07x2+24xy−34x+24y−185=0​

Answer 1

Rotating the graph of $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ with $B\ne0$ counterclockwise by $\theta$ eliminates the $xy$ term, where $\cot2\theta=\frac{A-C}{B}.$ Plugging in, we have $\cot2\theta=\frac{7}{24},$ since $C=0.$ Solving, we have $\theta=\frac{1}{2}\cot^{-1}(\frac{7}{24})\approx\boxed{36.9^\circ}.$