Question

if multiplicative inverse of 3 - 4i is x + iy , then the value of x and y are​

Answer 1

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Answer 2

SOLUTION

GIVEN

The multiplicative inverse of 3 - 4i is x + iy

TO DETERMINE

The value of x and y

CONCEPT TO BE IMPLEMENTED

Complex Number

A complex number z = a + ib is defined as an ordered pair of Real numbers ( a, b) that satisfies the following conditions :

(i) Condition for equality :

(a, b) = (c, d) if and only if a = c, b = d

(ii) Definition of addition :

(a, b) + (c, d) = (a+c, b+ d)

(iii) Definition of multiplication :

(a, b). (c, d) = (ac-bd , ad+bc )

Of the ordered pair (a, b) the first component a is called Real part of z and the second component b is called Imaginary part of z

EVALUATION

The multiplicative inverse of 3 - 4i

$\displaystyle \sf{ = \frac{1}{3 - 4i} }$

$\displaystyle \sf{ = \frac{3 + 4i}{(3 + 4i)(3 - 4i)} }$

$\displaystyle \sf{ = \frac{3 + 4i}{{(3)}^{2} - {(4i)}^{2} } }$

$\displaystyle \sf{ = \frac{3 + 4i}{9 - 16 {i}^{2} } }$

$\displaystyle \sf{ = \frac{3 + 4i}{9 + 16 } }$

$\displaystyle \sf{ = \frac{3 + 4i}{25 } }$

$\displaystyle \sf{ = \frac{3 }{25 } + i \frac{4}{25} }$

So by the given condition

$\displaystyle \sf{ x + iy= \frac{3 }{25 } + i \frac{4}{25} }$

$\displaystyle \sf{ \implies \: x = \frac{3 }{25 } \: \: and \: \: y = \frac{4}{25} }$

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