F=52x^2-72x*y+73y^2-280x+290y+325

ContourPlot[F==0, {x, -4,4}, {y, -4, 4}]

G=F/.{x->(a*X-b*Y+p), y->b*X+a*Y+q}

G=Expand[G]

res=Solve[{Coefficient[G, X]==0,Coefficient[G, Y]==0}, {p,q}]

Solve[(Coefficient[G,X*Y]/.{a->1})==0, b]

a=3/5

b=-4/5

G

G/.res

360*(2*Pi-N[ArcCos[3/5]])/(2*Pi)

360*