Question

Which of the following of x and y would make up the following expression represent a real number? (4+5i)(x+yi)

Answer 1

x = 4 and y = - 5

Explanation:

note that the product of a complex number and it's conjugate is real

That is

(a + bi)(a - bi) where a, b are real

= a² - abi + abi - b²i²

= a² + b² ← a real number

For (4 + 5i)(x + yi) to be real

we require (x + yi) to be the conjugate of 4 + 5i , that is 4 - 5i

(4 + 5i)(4 - 5i) ⇒ x = 4 and y = - 5