If z1 and z2 are complex numbers such that z1 ≠ z2 and |z1|=|z2|. If z1 has positive real part and z2 has negative imaginary part, then ((z1+z2)/(z1-z2)) may be

Question

If z1 and z2 are complex numbers such that z1 ≠ z2 and |z1|=|z2|. If z1 has positive real part and z2 has negative imaginary part, then ((z1+z2)/(z1-z2)) may be (A) Purely imaginary (B) Real and positive (C) Real and negative (D) None of these

Tardigrade Admin · Accepted Answer

Let z1=a+i b=(a, b) and z2=c-i d=(c,-d) Where a>0 and d>0 Then |z1|=|z2| ⇒ a2+b2=c2+d2 Now (z1+z2/z1-z2)=((a+i b)+(c-i d)/(a+i b)-(c-i d)) =([(a+c)+i(b-d)][(a-c)-i(b+d)]/[(a-c)+i(b+d)][(a-c)-i(b+d)]) =((a2+b2)-(c2+d2)-2(a d+b c) i/a2+c2-2 a c+b2+d2+2 b d) =(-(a d+b c) i/a2+b2-a c+b d) [using (1)] Hence, ((z1+z2)/(z1-z2)) is purely imaginary. However if a d+b c=0, then ((z1+z2)/(z1-z2)) will be equal to zero. According to the conditions of the equation, we can have a d+b c=0

Q. If z1​ and z2​ are complex numbers such that z1​=z2​ and ∣z1​∣=∣z2​∣. If z1​ has positive real part and z2​ has negative imaginary part, then (z1​−z2​)(z1​+z2​)​ may be

Purely imaginary

Real and positive

Real and negative

None of these

Q. If $z_{1}$ and $z_{2}$ are complex numbers such that $z_{1} \neq = z_{2}$ and $∣ z_{1} ∣ = ∣ z_{2} ∣$ . If $z_{1}$ has positive real part and $z_{2}$ has negative imaginary part, then $\frac{( z _{1} + z _{2} )}{( z _{1} - z _{2} )}$ may be