Question

The reflection of the complex number $\dfrac{{4 + 3i}}{{1 + 2i}}$ in the straight line $iz = \overline z$, is
(A) $2 + i$
(B) $2 - i$
(C) $1 + 2i$
(D) $1 - 2i$

Answer 1

We have given a complex number in the straight line $iz = \overline z$. We have to find the reflection of that complex number. We have to find the coordinates of this complex number in the line. This can be found by doing realization of the complex number. After that we will find the equation of the line in $x$ and $y$ form. This representation gives us the reflection of the complex number.

Complete step-by-step answer:
We have given the complex number $\dfrac{{4 + 3i}}{{1 + 2i}}$ in the line $iz = \overline z$.
Let us put $\dfrac{{4 + 3i}}{{1 + 2i}} = {z_1}$
So ${z_1} = \dfrac{{4 + 3i}}{{1 + 2i}}$
We have to find the coordinates of the point.
So multiplying and diving ${z_1}$ by $1 - 2i$
${z_1} = \dfrac{{4 + 3i}}{{1 + 2i}} \times \dfrac{{1 - 2i}}{{1 - 2i}} = \dfrac{{(4 + 3i)(1 - 2i)}}{{(1 \times 2i)(1 - 2i)}}$
$= \dfrac{{4 - 8i + 3i - 6{i^2}}}{{{{(1)}^2} - {{(2i)}^2}}} = \dfrac{{4 - 5i - 6( - 1)}}{{1 - 2( - 1)}}$
$= \dfrac{{4 - 5i + 6}}{{1 + 4}} = \dfrac{{10 - 5i}}{5} = \dfrac{{5(2 - i)}}{5}$
$= 2 - i$
So, $z = 2 - i$
Coordinate of ${x_1}$ is $(2, - 1)$
Now we have given the line $iz = \overline z$
$\Rightarrow$ $iz - \overline z = 0$
$\Rightarrow$ $i(x + iy) = (x - iy) = 0$ because $z = x + iy$
$\Rightarrow$ $ix + {i^2}y - x + iy = 0$
$\Rightarrow$ $ix - y - x + iy = 0$
$\Rightarrow$ $i(x + y) - 1(x + y) = 0$
$\Rightarrow$ $(x + y)(i = 1) = 0$
This represents the line $y = - x$.

Hence reflection of the point $(2, - 1)$ in the line $y = - x$ lines the point $(1, - 2)$ which is equal to $1 - 2i$ in the wig and plant.

Note: The complex numbers in mathematics are those numbers which can be written in $a + ib$ form where $'L'$ is the imaginary number called iola and has the value $\sqrt { - 1}$.
e.g. $2 + 3i$ is a complex number where $2$ is its real part and $3i$ is its imaginary part. The combinations of both real and imaginary parts are called complex numbers. The main application of these numbers is that they represent periodic motion such as water waves, alternating current, Light waves etc.
There are four types of algebraic expression. Which we can apply to complex numbers. There operations are Addition, Subtraction, Multiplication and Division