Question

Which values of x and y would make the following expression represent a real number?

(4 + 5i)(x + yi)
x = 4, y = 5
x = –4, y = 0
x = 4, y = –5
x = 0, y = 5

Answer 1

We require the imaginary part of (4+5i)(x+yi) to be 0.

(4+5i)(x+yi) = (4x − 5y) + i(5x + 4y), so 5x + 4y = 0.

Checking options:
- (4,5): 5·4 + 4·5 = 40 ≠ 0
- (−4,0): 5·(−4) + 4·0 = −20 ≠ 0
- (4,−5): 5·4 + 4·(−5) = 20 − 20 = 0 ✓
- (0,5): 5·0 + 4·5 = 20 ≠ 0

Thus x = 4, y = −5. (The product equals 41 in that case.)