Question

Show that the perpendicular distance of a point d (d is a complex number) on the Argand plane from the line \[\overline a z + a\overline z + b = 0\] (a is a complex number and b is a real) is \[\dfrac{{\left. {\left| {a\overline d + \overline a d + b} \right.} \right|}}{{2\left| {\left. a \right|} \right.}}\] .

Answer 1

Hint: In order to solve the question given above, you need to know about complex numbers. They refer to numbers that can be expressed in the form \[x + iy\] , where x and y are real numbers and i is an imaginary unit. You also need to understand the concept of complex plane or argand plane, which refers to a plot of complex numbers as points.

Complete step by step solution:
We know that the equation of the line is: \[\overline a z + a\overline z + b = 0\] .
Now, we have to put \[z = x + iy\] .
The above equation becomes:
 \[a\left( {x - iy} \right) + \overline a \left( {x + iy} \right) + b = 0\] .
Therefore, from the above equation, we get that:
 \[\left( {a + \overline a } \right)x + \left( {\overline a - a} \right)iy + b = 0\] .
Now, let \[d = {d_1} + i{d_2}\] in the equation,
With the help of the above information, you can calculate the distance of the line from d:
 \[\dfrac{{\left. {\left| {\left( {a + \overline a } \right)} \right.{d_1} + \left( {\overline a - a} \right)i{d_2} + b} \right|}}{{\sqrt {{{\left( {a + \overline a } \right)}^2}} + {{\left[ {\left( {\overline a - a} \right)i} \right] }^2}}}\] .
This can be written as:
 \[\dfrac{{\left. {\left| {a\left( {{d_1} - i{d_2}} \right) + \overline a \left( {{d_1} + i{d_2}} \right)} \right. + b} \right|}}{{\sqrt {{{\left( {a + \overline a } \right)}^2} - {{\left( {\overline a - a} \right)}^2}} }}\]
 \[ \Rightarrow \dfrac{{\left. {\left| {a\overline d + \overline a d + b} \right.} \right|}}{{\sqrt {4\overline a a} }}\] .
This gives us:
 \[\dfrac{{\left. {\left| {a\overline d + \overline a d + b} \right.} \right|}}{{\sqrt {4\left| {{{\left. a \right|}^2}} \right.} }}\]
 \[ \Rightarrow \dfrac{{\left. {\left| {a\overline d + \overline a d + b} \right.} \right|}}{{2\left| {\left. a \right|} \right.}}\] , which is our required answer.
Hence, proved.

Note: To solve questions similar to the one given above, you need to have basic understanding about a few important topics. These topics are: 1) complex numbers: they are numbers which include both the real and the imaginary parts. They can be expressed in the form \[x + iy\] , where x and y are real numbers and i is an imaginary unit and 2) argand plane: it refers to a diagram which contains the plot of complex numbers as points.