Question

Consider the following.1. zz¯=|z|22. z−1=z|z|2, where z= complex numberWhich of the above statement is/are correct?

• A. (A) Only 1
• B. (B) Only 2
• C. (C) Both 1 and 2
• D. (D) Neither 1 nor 2

Answer 1

Answer: (A)

(A) Only 1

Answer 2

Explanation:
Let, Complex number, z=a+ibConjugate of complex number =z¯=a−ibSo, zz¯=(a+ib)(a−ib)⇒zz¯=a2−(ib)2⇒zz¯=a2−i2(b)2⇒zz¯=a2−(ib)2⇒zz¯=a2+b2...(1) (∵i2=−1)As we know,Modulus of complex number is given by, |z|=(a2+b2)On squaring both sides, we get|z|2=(a2+b2)2∴|z|2=a2+b2...(2)Comparing equation (1) and (2), we getzz¯=|z|2Now,z−1=1z=1a+ib⇒z−1=1a+ib×a−iba−ib⇒z−1=1a+ib×a−iba−ib⇒z−1=a−iba2−(ib)2⇒z−1=a−iba2−i2b2⇒z−1=a−iba2−(−1)b2[∵i2=−1]⇒z−1=a−iba2+b2∴z−1=z¯|z|2≠z|z|2So, only statement 1 is correct.Hence, the correct option is (A).