Question

If \[Z = \dfrac{{\left( {4 + 3i} \right)}}{{\left( {5 - 3i} \right)}}\] then \[{Z^{ - 1}}\] equals to
A) \[\dfrac{{11}}{{25}} - \dfrac{{27}}{{25}}i\]
B) \[ - \dfrac{{11}}{{25}} - \dfrac{{27}}{{25}}i\]
C) \[ - \dfrac{{11}}{{25}} + \dfrac{{27}}{{25}}i\]
D) \[\dfrac{{11}}{{25}} + \dfrac{{27}}{{25}}i\]

Answer 1

Hint: Here in this question given a complex number, we have to find the simplest form of their inverse or reciprocal. For this first we need to write the reciprocal of a given complex number by interchanging the numerator and denominator and further simplify by multiply and divide by conjugate of the denominator of the resultant complex number to get the required solution.

Complete step by step answer:
A complex number generally denoted as Capital Z \[\left( Z \right)\] is any number that can be written in the form \[x + iy\] it’s always represented in binomial form. Where, \[x\] and \[y\] are real numbers. ‘\[x\]’ is called the real part of the complex number, ‘\[y\]’ is called the imaginary part of the complex number, and ‘\[i\]’ (iota) is called the imaginary unit.
Consider, the given complex number
\[Z = \dfrac{{\left( {4 + 3i} \right)}}{{\left( {5 - 3i} \right)}}\] -----(1)
We need to find the simplest form of reciprocal or inverse of complex number \[{Z^{ - 1}}\].
First, we need to reciprocal of equation (1) by interchanging the numerator and denominator, then we have
\[ \Rightarrow \,\,\dfrac{1}{Z} = \dfrac{{\left( {5 - 3i} \right)}}{{\left( {4 + 3i} \right)}}\]
\[ \Rightarrow \,\,{Z^{ - 1}} = \dfrac{{\left( {5 - 3i} \right)}}{{\left( {4 + 3i} \right)}}\] -----(2)
Multiply and divide by conjugate of the denominator \[\left( {4 - 3i} \right)\], then
\[ \Rightarrow \,\,{Z^{ - 1}} = \dfrac{{\left( {5 - 3i} \right)}}{{\left( {4 + 3i} \right)}} \times \dfrac{{\left( {4 - 3i} \right)}}{{\left( {4 - 3i} \right)}}\]
\[ \Rightarrow \,\,{Z^{ - 1}} = \dfrac{{\left( {5 - 3i} \right)\left( {4 - 3i} \right)}}{{\left( {4 + 3i} \right)\left( {4 - 3i} \right)}}\]
Now apply a algebraic identity \[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\] in denominator, then we have
\[ \Rightarrow \,\,{Z^{ - 1}} = \dfrac{{\left( {5 - 3i} \right)\left( {4 - 3i} \right)}}{{{{\left( 4 \right)}^2} - {{\left( {3i} \right)}^2}}}\]
\[ \Rightarrow \,\,{Z^{ - 1}} = \dfrac{{20 - 15i - 12i + 9{{\left( i \right)}^2}}}{{16 - 9{{\left( i \right)}^2}}}\]
As we know the value of \[i = \sqrt { - 1} \] and \[{i^2} = - 1\], then
\[ \Rightarrow \,\,{Z^{ - 1}} = \dfrac{{20 - 15i - 12i + 9\left( { - 1} \right)}}{{16 - 9\left( { - 1} \right)}}\]
\[ \Rightarrow \,\,{Z^{ - 1}} = \dfrac{{20 - 15i - 12i - 9}}{{16 + 9}}\]
On simplification, we get
\[ \Rightarrow \,\,{Z^{ - 1}} = \dfrac{{11 - 27i}}{{25}}\]
Or
\[\therefore \,\,{Z^{ - 1}} = \dfrac{{11}}{{25}} - \dfrac{{27}}{{25}}i\]. Therefore, option (A) is the correct answer.

Note:
The complex number is in the form of \[Z = x + iy\] and \[{Z^{ - 1}}\] is represented as a reciprocal or inverse of complex numbers. Remember the conjugate of a complex number Z bar (\[\overline Z \]) is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign and must know the basic algebraic identities.