Write the appropriate rotation formulas so that in a rotated system the equation has no x'y'-term. 10x2 - 4xy + 6y2 - 8x + 8y = 0

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)