If 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 simplify the expression z1/z2.
Given:
Z1 = 1 - i
Z2 = 3 - ai
z1/z2 = 5/13 - 1/13i
Now, let's express z1 and z2 in terms of their real and imaginary parts:
Z1 = x1 + y1i
Z2 = x2 + y2i
where x1, y1, x2, and y2 are real numbers.
From the given information, we know that:
x1 = 1, y1 = -1
x2 = 3, y2 = -a
Now, let's write the expression for z1/z2 and simplify it:
z1/z2 = (x1 + y1i)/(x2 + y2i)
= (1 - i)/(3 - ai)
= (1 - i)/(3*(1 - ai/3))
= (1 - i)/(3 - ai)
= ((1 - i)/(3 - ai)) * ((3 + ai)/(3 + ai))
= ((3 + 3i - 3i - i^2)/(9 - 3ai + 3ai - a^2i^2))
= ((3 - 1 + 3i)/9 - a^2)
= (2 + 3i)/(9 - a^2)
= 5/13 - 1/13i
Now, we can equate the real and imaginary parts of both expressions to find the value of a:
Real part: 2/(9 - a^2) = 5/13
Imaginary part: 3/(9 - a^2) = -1/13
From the real part:
2/(9 - a^2) = 5/13
26 = 45 - 5a^2
5a^2 = 19
a^2 = 19/5
a = ±√(19/5)
Since a is a real number, we will take the positive square root:
a = √(19/5)