I am required to calculate the rotation angle needed to come into standard form without x y product term (to make axes parallel to conic axes) in trying to find solution of problem:
A conic $M$, in standard or reflected standard form, is rotated through an angle $r$ about the origin to obtain the conic $N$ with equation $ax^2+bxy+cy^2=d$ where $a=2$, $b=2$, $c=34$ and $d=54$.
To two decimal places what is the absolute value of the angle $r$ in degrees? As usual, do not give any units in your answer. Do not include a minus sign in your answer.
This is the solution I came up with: $2x^2+2xy+34y^2=54$ $A=2 \quad B=2 \quad C=34 \quad D=54$
$A \neq C, $ \begin{align*} \therefore \theta &= \tfrac{1}{2} \tan^{-1} \tfrac{2}{2-34} \\ &= \tfrac{1}{2} (-0.06242) \\ &= -0.03121 \\ &\simeq -0.03 \end{align*} Someone please check my work. Thanks.