Let z₀ = 3 + 4i be a fixed point in the complex plane. The distance of a complex number from z₀ is 5 , and the distance of the complex number 4 from z₀ is 29 . Then the value of 10| |^2 is ________.

Step-by-step solution

The distance of from z₀ is 5 , which means: | - z₀| = 5 | - z₀|^2 = 25 Expanding this squared modulus: ( - z₀)( - z ₀) = 25 | |^2 - z ₀ - z₀ + |z₀|^2 = 25 Since z₀ = 3 + 4i , we have |z₀|^2 = 3^2 + 4^2 = 25 . Substituting this into the expansion: | |^2 - ( z ₀ + z₀) + 25 = 25 z ₀ + z₀ = | |^2 The distance of 4 from z₀ is 29 , which means: | 4 - z₀ |^2 = 29 Expanding this squared modulus: ( 4 - z₀ ) ( 4 - z ₀ ) = 29 16 | |^2 - 4 z ₀ - 4z₀ + |z₀|^2 = 29 16 | |^2 - 4( z ₀ + z₀) | |^2 + 25 = 29 Substitute z ₀ + z₀ = |