JEE Mains · Maths · STD 11 - 4.1 complex nubers
For two non-zero complex number \(z_1\) and \(z_2\), if \(\operatorname{Re}\left(z_1 z_2\right)=0\) and \(\operatorname{Re}\left(z_1+z_2\right)=0\), then which of the following are possible ? \((A)\) \(\operatorname{Im}\left(z_1\right) > 0\) and \(\operatorname{Im}\left(z_2\right) > 0\) \((B)\) \(\operatorname{Im}\left(z_1\right) < 0\) and \(\operatorname{Im}\left(z_2\right) > 0\) \((C)\) \(\operatorname{Im}\left(z_1\right) > 0\) and \(\operatorname{Im}\left(z_2\right) < 0\) \((D)\) \(\operatorname{Im}\left( z _1\right) < 0\) and \(\operatorname{Im}\left( z _2\right) < 0\) Choose the correct answer from the options given below :
- A \(B\) and \(D\)
- B \(B\) and \(C\)
- C \(A\) and \(B\)
- D \(A\) and \(C\)
Answer & Solution
Correct Answer
(B) \(B\) and \(C\)
Step-by-step Solution
Detailed explanation
\(z _1= x _1+ i y _1\) \(z _2= x _2+ iy _2\) \(\operatorname{Re}\left(z_1 z_2\right)=x_1 x_2-y_1 y_2=0\) \(\operatorname{Re}\left(z_1+z_2\right)=x_1+x_2=0\) \(x_1\) and \(x_2\) are of opposite sign \(y_1\) and \(y_2\) are of opposite sign
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