Question

If (3-4i) (x+yi) = 1 + 0i. The value of x and y is
(a). (3/25,4/25)
(b). (1/6,0)
(c). (1,1)
(d). (0.5)​

Answer 1

Step-by-step explanation:

$\textsf{\large{\underline{Solution}:}}$

Given That:

$\rm: \longmapsto (3 - 4i)(x + iy) = 1 + 0i$

$\rm: \longmapsto 3(x + iy) - 4i(x + iy)= 1 + 0i$

$\rm: \longmapsto 3x + 3iy - 4ix +4y= 1 + 0i$

On rearranging the terms, we get:

$\rm: \longmapsto (3x + 4y)+ (3y - 4x)i = 1 + 0i$

Comparing both sides, we get:

$\rm: \longmapsto 3x + 4y = 1 --- (i)$

$\rm: \longmapsto 3y - 4x = 0 --- (ii)$

Multiplying (i) by 4, we get:

$\rm: \longmapsto 12x + 16y = 4--- (iii)$

Multiplying (ii) by 3, we get:

$\rm: \longmapsto 9y - 12x = 0 --- (iv)$

Adding equations (iii) and (iv), we get:

$\rm: \longmapsto25y = 4$

$\rm: \longmapsto y =\dfrac{4}{25}$

From (ii), we get:

$\rm: \longmapsto 3y - 4x = 0$

$\rm: \longmapsto 3y = 4x$

$\rm: \longmapsto x = \dfrac{3}{4} y$

$\rm: \longmapsto x = \dfrac{3}{4} \times \dfrac{4}{25}$

$\rm: \longmapsto x = \dfrac{3}{25}$

Therefore:

$\rm: \longmapsto (x,y)= \bigg( \dfrac{3}{25}, \dfrac{4}{25} \bigg)$

★ Which is our required answer.

$\textsf{\large{\underline{More To Know}:}}$

$\rm1.\ i^{4n} = 1$

$\rm2. \ i^{4n+1} = i$

$\rm3.\ i^{4n+2} = -1$

$\rm4.\ i^{4n+3} = -i$