Determine the value of a if (surd3-ai)over(1-surd3i) is a real no.& find this real no..?

Question

Determine the value of a if (surd3-ai)over(1-surd3i) is a real no.& find this real no..?

Answer 1

Let s = [ sqrt(3) - ai ] / [ 1 - i sqrt(3)  ]

Multiplying top and bottom by 1 + i sqrt(3):

s = [ (sqrt(3) - ai)(1 + i sqrt(3)) ] / (1 + 3)

= [ sqrt(3) - ai + 3i + a sqrt(3) ] /  4

= [ (1 + a)sqrt(3) + i(3 - a) ] / 4.

s is real if a = 3, and its value is then (1 + a) sqrt(3) / 4.