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)