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Question: If \[Z = \dfrac{{\left( {4 + 3i} \right)}}{{\left( {5 - 3i} \right)}}\] then \[{Z^{ - 1}}\] equals t...

If Z=(4+3i)(53i)Z = \dfrac{{\left( {4 + 3i} \right)}}{{\left( {5 - 3i} \right)}} then Z1{Z^{ - 1}} equals to
A) 11252725i\dfrac{{11}}{{25}} - \dfrac{{27}}{{25}}i
B) 11252725i - \dfrac{{11}}{{25}} - \dfrac{{27}}{{25}}i
C) 1125+2725i - \dfrac{{11}}{{25}} + \dfrac{{27}}{{25}}i
D) 1125+2725i\dfrac{{11}}{{25}} + \dfrac{{27}}{{25}}i

Explanation

Solution

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 (Z)\left( Z \right) is any number that can be written in the form x+iyx + iy it’s always represented in binomial form. Where, xx and yy are real numbers. ‘xx’ is called the real part of the complex number, ‘yy’ is called the imaginary part of the complex number, and ‘ii’ (iota) is called the imaginary unit.
Consider, the given complex number
Z=(4+3i)(53i)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 Z1{Z^{ - 1}}.
First, we need to reciprocal of equation (1) by interchanging the numerator and denominator, then we have
1Z=(53i)(4+3i)\Rightarrow \,\,\dfrac{1}{Z} = \dfrac{{\left( {5 - 3i} \right)}}{{\left( {4 + 3i} \right)}}
Z1=(53i)(4+3i)\Rightarrow \,\,{Z^{ - 1}} = \dfrac{{\left( {5 - 3i} \right)}}{{\left( {4 + 3i} \right)}} -----(2)
Multiply and divide by conjugate of the denominator (43i)\left( {4 - 3i} \right), then
Z1=(53i)(4+3i)×(43i)(43i)\Rightarrow \,\,{Z^{ - 1}} = \dfrac{{\left( {5 - 3i} \right)}}{{\left( {4 + 3i} \right)}} \times \dfrac{{\left( {4 - 3i} \right)}}{{\left( {4 - 3i} \right)}}
Z1=(53i)(43i)(4+3i)(43i)\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 a2b2=(a+b)(ab){a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right) in denominator, then we have
Z1=(53i)(43i)(4)2(3i)2\Rightarrow \,\,{Z^{ - 1}} = \dfrac{{\left( {5 - 3i} \right)\left( {4 - 3i} \right)}}{{{{\left( 4 \right)}^2} - {{\left( {3i} \right)}^2}}}
Z1=2015i12i+9(i)2169(i)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=1i = \sqrt { - 1} and i2=1{i^2} = - 1, then
Z1=2015i12i+9(1)169(1)\Rightarrow \,\,{Z^{ - 1}} = \dfrac{{20 - 15i - 12i + 9\left( { - 1} \right)}}{{16 - 9\left( { - 1} \right)}}
Z1=2015i12i916+9\Rightarrow \,\,{Z^{ - 1}} = \dfrac{{20 - 15i - 12i - 9}}{{16 + 9}}
On simplification, we get
Z1=1127i25\Rightarrow \,\,{Z^{ - 1}} = \dfrac{{11 - 27i}}{{25}}
Or
Z1=11252725i\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+iyZ = x + iy and Z1{Z^{ - 1}} is represented as a reciprocal or inverse of complex numbers. Remember the conjugate of a complex number Z bar (Z\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.