Given that Z1=1-i and Z2=3-ai
Find the value of a if Z1/Z2 =5/13-1/13i
To find the value of a, we first need to calculate Z1/Z2 and then equate it to 5/13 - 1/13i.
Z1/Z2 = (1 - i) / (3 - ai)
=> (1 - i) * (3 + ai) / [(3 - ai) * (3 + ai)]
=> (3 + 3i + ai - 3i) / (9 - 3ai + 3ai - a^2)
=> (3 + ai) / (9 - a^2)
Now, we equate it to 5/13 - 1/13i:
(3 + ai) / (9 - a^2) = 5/13 - 1/13i
Solving this, we get:
3 + ai = 5(9 - a^2) / 13 - 1(9 - a^2) / 13
=> 3 + ai = (45 - 5a^2) / 13 - (9 - a^2) / 13
=> 3 + ai = (45 - 5a^2 - 9 + a^2) / 13
=> 3 + ai = (36 - 4a^2) / 13
Multiplying by 13 to both sides:
39 + 13ai = 36 - 4a^2
=> 13ai + 4a^2 = 3
Comparing real and imaginary parts:
- 4a^2 = 3
=> a = -√(3/4) or a = √(3/4)
Hence, the value of a is a = -√(3/4) or a = √(3/4).