Question

If $\overline{z}$ lies in the third quadrant then $z$ lies in the
$A.$ First quadrant
$B.$ Second quadrant
$C.$Third quadrant
$D.$ Fourth quadrant

Answer 1

Hint: This question can be solved by comparing the general value of $\overline{z}$ and $\overline{z}$when it is in the third quadrant.

Now we know that the general value of $z=x+iy$
And $\bar{z}=x-iy-----\left(i\right)$
Now given that $\overline{z}$ lies in the third quadrant.
$\Rightarrow \bar{z}=-x-iy------\left(ii\right)$
Where the negative sign indicates that both the real part and imaginary part lies in the third quadrant.
On comparing $\left(i\right)$ and$\left(ii\right)$we get,
$x=-x$
Also we know that the general value of $z=x+iy$
Putting the value of $x$ in general value of $z$ we get,
$z=-x+iy$
On analyzing the above equation we can say that $z$ is in the Second quadrant because here $\left(x\right)$ coordinate is negative and$\left(y\right)$ coordinate is positive.
$\therefore$ The correct answer is $\left(B\right)$.

Note: Whenever we face such type of questions the key concept is that we should compare the given value of $\overline{z}$and general value of $\overline{z}$ so we can compare both the equations and we get the value of $x$ and we also know the general value of $z$ and on putting the value of $x$ in it we get the position of $z$.