Question

Two complex numbers Z1 = a+ib , Z2 = c+id are equal if

a) a=c , b= d
b) a=c
c) b=d
d) a=b , b= id

pls explain the answer and the crct and best answer will be marked as brainliest for sure ​

Answer 1

Answer:

ANSWER

Let, z

1

=a+ib and z

2

=c+id such that ∣z

1

∣=1⇒

a

2

+b

2

=1⇒a

2

+b

2

=1......(1)

and ∣z

2

∣=1⇒

c

2

+d

2

=1⇒c

2

+d

2

=1......(2).

Also, Re(z

1

z

2

)=0

or, Re{ac+bd+i(bc−ad)}=0

or, ac+bd=0

or, a=−

c

bd

.....(3).

Using (3) in (1) we get,

c

2

b

2

d

2

+b

2

=1

or,

c

2

b

2

(d

2

+c

2

)

=1

or, b

2

=c

2

[Using (2)]........(4).

Similarly using (3) in (2) and using (1) we shall have

a

2

=d

2

.........(5).

Now, according to the problem w

1

=a+ic and w

2

=b+id.

∴∣w

1

∣=

a

2

+c

2

=

a

2

+b

2

=1 [Using (4)].

∴∣w

2

∣=

b

2

+d

2

=

b

2

+a

2

=1 [Using (5)].

Now, w

1

w

2

=(a+ic)(b−id)=(ab+cd)+i(bc−ad).

∴Re(w

1

w

2

)=ab+cd=−

c

b

2

d

+cd=

c

d(c

2

−b

2

)

=0 [Using (3) and (4)].

So, option (A), (B) and (C) are true