For two non-zero complex number z 1 and z 2 , if Re z 1 z 2 = 0 and Re z 1 + z 2 = 0 , then which of the following are possible? (A) Im z 1 > 0 and Im z 2 > 0 (B) Im z 1 < 0 and Im z 2 > 0 (C) Im z 1 > 0 and Im z 2 < 0 (D) Im z 1 < 0 and Im z 2 < 0 Choose the…

Question

For two non-zero complex number z 1 and z 2 , if Re z 1 z 2 = 0 and Re z 1 + z 2 = 0 , then which of the following are possible? (A) Im z 1 &#62; 0 and Im z 2 &#62; 0 (B) Im z 1 &#60; 0 and Im z 2 &#62; 0 (C) Im z 1 &#62; 0 and Im z 2 &#60; 0 (D) Im z 1 &#60; 0 and Im z 2 &#60; 0 Choose the correct answer from the options given below:

Aicharya · Accepted Answer

Let, z 1 = x 1 + i y 1 and z 2 = x 2 + i y 2 &#8757; Re z 1 z 2 = 0 x 1 x 2 - y 1 y 2 = 0 . . . . . . i &#8757; Re z 1 + z 2 = 0 x 1 + x 2 = 0 . . . . . . i i From equations i and i i we get x 1 2 + y 1 y 2 = 0 &#8658; y 1 y 2 = - x 1 2 Therefore I m ( z 1 ) and I m ( z 2 ) are of opposite sign

For two non-zero complex number z 1 and z 2 , if Re z 1 z 2 = 0 and Re z 1 + z 2 = 0 , then which of the…

Solution