Find the area enclosed by the conic:

$\large{13x^2 + 13y^2 - 10xy -60 \sqrt{2} x + 12 \sqrt{2} y +72 = 0}$

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Details and Assumptions:
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- If you think that the conic is not a closed path, then type your answer as $0$ .

The answer is 18.8495559215.

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At first,rotate the the co-ordinate axes through $45$ degrees(anticlockwise) to get $\frac{(x'-3)^2}{9}+\frac{(y'+2)^2}{4}=1$ .

So the area is $6\pi$ .

In the above I used the following:

1.The transformation equation for rotaion of $θ$ is $x=x'cosθ-y'sinθ$ and $y=x'sinθ+y'cosθ$ .

2.I chose $θ$ to be $45$ degrees to eliminate the $xy$ term.