Two complex numbers Z1 and Z2 are such that z1 + z2=1
If z1=a/1+i and z2= b/1+2i. Find the values of a and b
Multiplying both sides by (1-i) will not rationalize the denominator of both fractions. You need to do:
a/(1+i) + b/(1+2i) = 1
a(1-i)/(1+1) + b(1-2i)/(1+4) = 1
5a(1-i) + 2b(1-2i) = 10
5a-5ai + 2b-4ai = 10+0i
5a+2b = 10
5a+4b = 0
(a,b) = (4,-5)
Sorry for the oversight in the previous response. Thank you for pointing it out.
Using the correct rationalization method as provided:
a/(1+i) + b/(1+2i) = 1
a(1-i)/2 + b(1-2i)/5 = 1
(5a(1-i) + 2b(1-2i))/10 = 1
5a - 5ai + 2b - 4bi = 10
(5a + 2b) + (-5a - 4b)i = 10
From the real and imaginary parts:
5a + 2b = 10
-5a - 4b = 0
Solving these simultaneous equations gives:
5a + 2b = 10
5a = 4b
a = 4/5, b = -5
Therefore, the values of a and b are 4/5 and -5, respectively.
Given that z1 + z2 = 1, we can write this as:
(a/1+i) + (b/1+2i) = 1
Multiplying both sides by (1-i) to rationalize the denominator:
(a(1-i) + b(1-2i)) = (1-i)
Expanding the left side:
a - ai + b - 2bi = 1 - i
Grouping the real parts and imaginary parts:
(a + b) + (-a - 2b)i = 1 - i
Now comparing the real and imaginary parts on both sides:
a + b = 1 (1)
-a - 2b = -1 (2)
From equation (1):
a = 1 - b
Substitute a into equation (2):
-(1 - b) - 2b = -1
-1 + b - 2b = -1
b = 1
Therefore, the value of b is 1. Substituting b back into a = 1 - b:
a = 1 - 1
a = 0
So, the values of a and b are 0 and 1, respectively.