Question

If $z$ is a complex number satisfying $z+3\overline{z}=5-6i$, then the value of $Im\left(\overline{z}\right)$ is

• A. $-3$
• B. $3$
• C. $\frac{1}{4}$
• D. $\frac{5}{4}$

Answer 1

The correct option is A $-3$

Given : $z+3\overline{z}=5-6i$
Let $z=a+ib\Rightarrow \overline{z}=a-ib$
Now, putting in the given equation, we get
$\Rightarrow a+ib+3(a-ib)=5-6i\Rightarrow 4a-2ib=5-6i\Rightarrow 4a=5,-2b=-6\Rightarrow a=\frac{5}{4},b=3\Rightarrow z=\frac{5}{4}+3i\Rightarrow \overline{z}=\frac{5}{4}-3i\therefore Im\left(\overline{z}\right)=-3$