# If $\bar z$ lies in the third quadrant then $z$ lies in the

$A.$ First quadrant

$B.$ Second quadrant

$C.$Third quadrant

$D.$ Fourth quadrant

Last updated date: 26th Mar 2023

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Answer

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

Now we know that the general value of $z = x + iy$

And $\overline z = x - iy - - - - - \left( i \right)$

Now given that $\bar z$ lies in the third quadrant.

$ \Rightarrow \overline 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 $\bar z$and general value of $\bar z$ 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 $z$.

Now we know that the general value of $z = x + iy$

And $\overline z = x - iy - - - - - \left( i \right)$

Now given that $\bar z$ lies in the third quadrant.

$ \Rightarrow \overline 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 $\bar z$and general value of $\bar z$ 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 $z$.

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