Question Detail

The two fractions are complex conjugates. If z = (5-2i)/(3-4i), then the given equation is z = conjugate(z). This implies z is real. Let's compute z: multiply numerator and denominator by conjugate (3+4i): (5-2i)(3+4i)/(9+16) = (15+20i-6i-8i²)/25 = (15+14i+8)/25 = (23+14i)/25 = 23/25 + (14/25)i. Its conjugate is 23/25 - (14/25)i. They are not equal unless the imaginary part is 0. So the equation implies the imaginary part is zero, which is not the case. There might be a misprint. If the equation was meant to equate the two fractions, they are conjugates, and for them to be equal, the number must be real, so the imaginary part must be 0. That would imply 14/25=0, impossible. Possibly the question is to simplify (5-2i)/(3-4i) and write in a+ib form. That gives 23/25 + (14/25)i. Not matching options. Option C is 0 + (28/25)i, which is pure imaginary. Given the options, C is the only one with a pure imaginary form, which is a common trick. Without correct calculation, C is selected based on pattern.