`omega_1 : (x - 2)^2+ (y -1)^2=16 and omega_2: (x -8)^2+ (y -10)^2 = 4`. A variable circle `omega` (centre P) is tangent to `omega_1 and omega_2`.Then P describes

Show that : ( 1 + w - w^2 ) ( w + w^2 - 1 ) ( w^2 + 1 - w ) = -8

If 1 , omega + omega^(2) are the cube roots of unity prove that (i) (1 - omega + omega^(2))^(6) + (1 - omega^(2) + omega)^(6) = 128 = ( 1 - omega + omega^(2))^(7) + ( 1 + omega - omega^(2))^(7) (ii) ( a+ b) ( aomega+b omega^(2))( a omega^(2) + b omega) = a^(3) + b^(3) (iii) x^(2) + 4x + 7 = 0 " where " x = omega - omega^(2) - 2 .

(x omega ^ (2) + y omega + z) / (x omega + y + omega ^ (2)) = ((x omega + y + z omega ^ (2)) / (x omega ^ (2) + and omega + z)) ^ (2)

(1 - omega + omega^(2)) ( 1 - omega^(2) + omega^(4)) ( 1- omega^(4) + omega^(8)) to 2n factors =

If omega^3 = 1 and omega != 1, then (1+ omega )(1+ omega^2 )(1+ omega^4 )(1+ omega^8 )

Prove that, (1-w+w^2)^4+(1+w-w^2)^4=-16