Let z₀ = 2 + 2i . A complex number and its reciprocal conjugate 1 lie on the circles |z - z₀|^2 = 10 and |z - z₀|^2 = k respectively. If | |^2 = 1 3 , then the value of k is ________.

Step-by-step solution

Given that lies on |z - z₀|^2 = 10 , we can expand this as: ( - z₀)( - z ₀) = 10 | |^2 - z ₀ - z₀ + |z₀|^2 = 10 We have z₀ = 2 + 2i , so |z₀|^2 = 2^2 + 2^2 = 8 . Substituting |z₀|^2 = 8 into the equation: | |^2 - ( z ₀ + z₀) + 8 = 10 z ₀ + z₀ = | |^2 - 2 It is also given that 1 lies on |z - z₀|^2 = k . Expanding this: ( 1 - z₀ ) ( 1 - z ₀ ) = k 1 | |^2 - z ₀ - z₀ + |z₀|^2 = k 1 | |^2 - z ₀ + z₀ | |^2 + 8 = k Substitute z ₀ + z₀ = | |^2 - 2 into the above equation: 1 | |^2 - | |^2 - 2 | |^2 + 8 = k 1 - (| |^2 - 2) |