Answer:
x = x' cos (π/8) + y' sin (π/8)
y = -x' sin (π/8) + y' cos (π/8)
Step-by-step explanation:
Canonical form of conics section:
A*x^2 + B*x + C*y^2 + D*y + E*x*y + F = 0
We want that in the rotated system the equation has no x'y'-term. To do this the rotated angle has to satisfy:
tan(2 θ) = E/(C - A)
The rotation formula when the coordinates system rotates an angle θ are:
x = x' cos θ + y' sin θ
y = -x' sin θ + y' cos θ
The conic section is: 10*x^2 - 4*x*y + 6*y^2 - 8*x + 8*y = 0, then:
A = 10
B = -8
C = 6
D = 8
E = -4
F = 0
So,
tan(2 θ) = -4/(6 - 10)
2 θ = tan^-1 (1)
θ = (π/4)*(1/2) = π/8
Finally,
x = x' cos (π/8) + y' sin (π/8)
y = -x' sin (π/8) + y' cos (π/8)
