If we have an equation Ax2+2Bxy+Cy2+2Dx+2Ey+F=0, we should convert it into a canonical equation Ax1~2+C1y~2+F1=0.
Let S=A+C,δ=∣∣ABBC∣∣,Δ=∣∣ABDBCEDEF∣∣. In our case A=57,B=73,C=43,F=−576. Therefore, S=100,δ=2304,Δ=−1327104.
Next, we solve the system
⎩⎨⎧A1+C1=S,A1C1=δ,A1C1F1=Δ.A1=36,C1=64,F1=−576orA1=64,C1=36,F1=−576.
So the canonical equation will have form
36x~2+64y~2=576,242(6x~)2+242(8y~)2=1,42x~2+32y~2=1. Another version of equation is
32x~2+42y~2=1.
It is an ellipse with semiaxes 4 and 3.
The center can be calculated from system
{Ax0+By0+D=0,Bx0+Cy0+E=0.{57x0+73y0+0=0,73x0+43y0+0=0.x0=0,y0=0.
The angle of shift of the axes is
tanα=BA1−A=−3,α=−60∘.
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