Question

If \[\dfrac{{{{\left| {(a + ib)} \right|}^2}}}{{(a - ib)}} - \dfrac{{{{\left| {(a - ib)} \right|}^2}}}{{(a + ib)}} = x + iy\] then find the value of \[x\].

Answer 1

Hint: The given question is based on complex numbers. It can be solved easily by applying the following property for complex numbers:
\[z.\overline z = {\left| z \right|^2}\]
We have to expand the equation using the given property and then simplify to arrive at the value of \[x\].

Complete step by step solution:
Complex numbers are the numbers that are expressed in the form of \[a + ib\] where, \[a\],\[b\] are real numbers and ‘\[i\]’ is an imaginary number called “iota”. For example, \[4 + 6i\] is a complex number, where \[4\] is a real number (Re) and \[6i\] is an imaginary number (Im).
An imaginary number is usually represented by ‘\[i\]’ or ‘\[j\]’, which is equal to \[\sqrt { - 1} \]. Therefore, the square of the imaginary number gives a negative value.
Let \[z = a + ib\] be a complex number.
The Modulus of z is represented by \[\left| z \right|\].
Mathematically, \[\left| z \right| = \sqrt {{a^2} + {b^2}} \]
The conjugate of “\[z\]” is denoted by \[\overline z \].
Mathematically, \[\overline z = a - ib\]
We can solve the given equation as follows:
\[\dfrac{{{{\left| {(a + ib)} \right|}^2}}}{{(a - ib)}} - \dfrac{{{{\left| {(a - ib)} \right|}^2}}}{{(a + ib)}} = x + iy\]
Using the property \[z.\overline z = {\left| z \right|^2}\], we get,
\[\dfrac{{(a + ib)\overline {(a + ib)} }}{{(a - ib)}} - \dfrac{{(a - ib)\overline {(a - ib)} }}{{(a + ib)}} = x + iy\]
Converting the sign as per conjugate rule \[\overline z = a - ib\], we get,
\[\dfrac{{(a + ib)(a - ib)}}{{(a - ib)}} - \dfrac{{(a - ib)(a + ib)}}{{(a + ib)}} = x + iy\]
Simplifying the equation by dividing, we get,
\[(a + ib) - (a - ib) = x + iy\]
Opening the brackets, we get,
\[a + ib - a + ib = x + iy\]
\[2ib = x + iy\]
Writing the equation in form of complex number \[z = a + ib\], we get,
\[x + iy = 0 + i(2b)\]
Hence, from the above equation, we can conclude that the value of \[x\] corresponds to \[0\].

Therefore, \[x = 0\].

Note:
The complex number is a mixture of a real number and an imaginary number. The main application of these numbers is to represent periodic motions such as water waves, alternating current, light waves, etc., which rely on sine or cosine waves, etc.