Let the set of all values of k R such that the equation |z|^2 + (1-i)z + k(c+i) = 0 , z C , has at least one solution, be the interval [ , ] , where c is a real constant. If ( - )^2 = 40 , then the value of c^2 is:

Step-by-step solution

Let z = x + iy . Then |z|^2 = x^2 + y^2 . Substitute z into the given equation: x^2 + y^2 + (1-i)(x+iy) + k(c+i) = 0 x^2 + y^2 + x + iy - ix + y + kc + ik = 0 Separating the real and imaginary parts, we get: Real part: x^2 + y^2 + x + y + kc = 0 Imaginary part: y - x + k = 0 y = x - k Substitute y into the real part equation: x^2 + (x-k)^2 + x + (x-k) + kc = 0 x^2 + x^2 - 2kx + k^2 + 2x - k + kc = 0 2x^2 + 2(1-k)x + k^2 + k(c-1) = 0 For the equation to have at least one solution z C , there must be at least one rea