Let for some real numbers α and β , a = α - i β . If the system of equations 4 i x + 1 + i y = 0 and 8 cos 2 π 3 + i sin 2 π 3 x + a ¯ y = 0 has more than one solution then α β is equal to

Step-by-step solution

The given system of equations has more than one solution, then it must have infinitely many solutions. So, 4 i 8 cos 2 π 3 + i sin 2 π 3 = 1 + i α + i β ∵ a = α - i β ⇒ a = α + i β ⇒ 4 i 8 - 1 2 + i 3 2 = 1 + i α + i β ⇒ α i - β = - 1 - i + 3 i - 3 ⇒ - β + α i = - 1 - 3 + - 1 + 3 i ⇒ α = - 1 + 3 & - β = - 1 - 3 ∴ α β = - 1 + 3 1 + 3 = - 1 + 3 1 + 3 × 1 - 3 1 - 3 = - 1 - 3 2 1 - 3