3)Find a and b ifi) a + 2b + 2ai = 4 + 6iii) (a-b) + (a+b)i = a + 5iiii) (a+b) (2 + i) = b + 1 + (10 + 2a)iiv) abi = 3a - b + 12iv)1/a+ib=3-2ivi) (a+ib) (1+i) =2+i
Answers
i)
a + 2b + 2ai = 4
(a + 2b) + 2ai = 4 + 0i
- Equating real and imaginary parts
a + 2b = 4 and 2ai = 0i
a + 2b = 4... .1 and 2a = 0
a = 0/2 = 0 ......2
- Substituting eq 2 in 1
•°• 0 + 2b = 4
2b = 4
b = 4/2 = 2
Answer. a = 0 and b = 2
ii)
(a - b) + (a +b)i = a + 5i
- Equating real and imaginary parts
•°• (a - b) = a and (a + b)i = 5i
a - b = a and a + b = 5 ....1
b = 0 ...2
- substituting 2 in 1
a + 0 = 5
a = 5
Answer. a = 5 and b = 0
iii)
(a + b) (2 + i) = b + 1 + (10 + 2a)i
2(a + b) + i(a + b) = b + 1 + (10 + 2a)i
- Equating real and imaginary parts
2(a + b) = b + 1 and (a + b)i = (10 + 2a)i
2a + 2b = b + 1 and a + b = 10 + 2a
2a + 2b - b = 1 and b = 10 + 2a - a
2a + b = 1 and a - b = -10
- Solving both the Equations.
- Adding the Equations
•°• 2a + b = 1
a - b = -10
3a = -9
a = -9/3 = -3
substituting a = -3 in a - b = -10
•°• -3 - b = -10
-3 + 10 = b
7 = b
Answer. a = -3 and b = 7
vi)
abi = 3a - b + 12i
0 + abi = 3a - b + 12i
- Equating real and imaginary parts
0 = 3a - b and abi = 12i
3a = b and ab = 12
a = b/3 ....1 and ab = 12 ....2
substituting eq 1 in 2
•°• (b/3) × b = 12
b²/3 = 12
b² = 12 × 3 = 36
b = 6
substituting b = 6 in eq 1
•°• a = 6/3 = 2
a = 2
Answer. a = 2 and b = 6
v)
1/(a + bi) = 3 - 2i
1 = (3 - 2i) (a + bi)
1 = a (3 - 2i) + bi(3 - 2i)
1 = 3a - 2ai + 3bi - 2bi² ....{i² = -1}
1 = 3a - 2ai + 3bi + 2b
1 = 3a + 2b + (-2a + 3b)i
1 + 0i = 3a + 2b + (-2a + 3b)i
- Equating real and imaginary parts
3a + 2b = 1 and (-2a + 3b)i = 0i
3a + 2b = 1 and -2a + 3b = 0
3a + 2b = 1 and 3b = 2a
3a + 2b = 1 ...1 and b = 2a/3 ......2
substituting eq 2 in 1
•°• 3a + 2 × 2a/3 = 1
3a + 4a/3 = 1
(9a + 4a)/3 = 1
13a = 3
a = 3/13
substituting a = 3/13 in eq 1
3/13 + 2b = 1
2b = 1 - 3/13
2b = 1 - 3/13
2b = (13 - 3)/13
2b = 10/13
b = 10/13 × 2 = 5/13
Answer. a = 3/13 and b = 5/13
vi)
(a + bi) (1 + i) = 2 + i
1(a + bi) + i(a + bi) = 2 + i
a + bi + ai + bi² = 2 + i
a + bi + ai - b = 2 + i
a - b + (a + b)i = 2 + i
- Equating real and imaginary parts
a - b = 2 and (a + b)i = i
a - b = 2 and a + b = 1
Adding both the Equations
•°• a - b = 2
a + b = 1
2a = 3
a = 2/3
substituting a = 2/3 in Equation a - b = 2
•°• 2/3 - b = 2
2/3 - 2 = b
(2 - 6)/3 = b
b = -4/3
Answer. a = 2/3 and b = -4/3