Question

If $|z - \overline z | + |z + \overline z | = 2,\,$then z lies on
A) a circle
B) a square
C) an ellipse
D) a line

Answer 1

The meaning of bar z ($\overline z$) here is that the complex number z in which we change the sign of the imaginary part. It can be from positive to negative or from negative to positive. In this question we have to assume the complex number z in the form of some variables and then we can do further calculations. Also, modulus is given in this question. That means we have to find the magnitude in this particular question.

Complete step by step answer:
In the given question,
$|z - \overline z | + |z +\overline z | = 2,\,$
Let, $z = x + iy$ be a complex number.
Therefore, the value of $\overline z = x - iy$
As we know, putting the bar on complex numbers means changing the sign of imaginary part of complex number.
Now, put the values of $z$ and $\overline z$ in the above equation.
$|\left( {x + iy} \right) + \left( {x - iy} \right)| + |\left( {x + iy} \right) - \left( {x - iy} \right)| = 2$
On calculation, we get
$|x + iy + x - iy| + |x + iy - x + iy| = 2$
$|2x| + |2iy| = 2$
As we know that
$|2x| = \sqrt {{{\left( {2x} \right)}^2}}$
$|2iy| = \sqrt {{{\left( {2y} \right)}^2}}$
On putting these values in equation, we get
$\sqrt {{{\left( {2x} \right)}^2}} + \sqrt {{{\left( {2y} \right)}^2}} = 2$
Taking square root in LHS,
$2x + 2y = 2$
On dividing the whole equation by $2$, we get
$x + y = 1$
Therefore, $x + y = 1$ is an equation of straight line. Hence, the correct option is option (D).

Note: The meaning of z bar in this question is the conjugate of complex numbers. The complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign of imaginary part. The product of a complex number and its complex conjugate is a real number.