If 1- i -1- i 3-1- i -1- i 3- x - iy- find x - y

If (1+i/1 i )^3 (1 i/1+i )^3=x+iy ⇒ ((1+i)(1+i)/(1 i)(1+i) )^3 ((1 i)(1 i)/(1+i)(1 i) )^3=x+iy [Rationalizing the denominator] ⇒ (1+2i 1/1+1 )^3 (1 2i 1/1+1 ...

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Byju's Answer

Standard XII

Mathematics

Equality of 2 Complex Numbers

If 1+ i /1- i...

Question

$If {(\frac{1 + i}{1 - i})}^{3} - {(\frac{1 - i}{1 + i})}^{3} = x + i y, find (x,y)$

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Solution

$If {(\frac{1 + i}{1 - i})}^{3} - {(\frac{1 - i}{1 + i})}^{3} = x + i y \Rightarrow {(\frac{(1 + i) (1 + i)}{(1 - i) (1 + i)})}^{3} - {(\frac{(1 - i) (1 - i)}{(1 + i) (1 - i)})}^{3} = x + i y [Rationalizing the denominator] \Rightarrow {(\frac{1 + 2 i - 1}{1 + 1})}^{3} - {(\frac{1 - 2 i - 1}{1 + 1})}^{3} = x + i y \Rightarrow {(\frac{2 i}{2})}^{3} - {(\frac{- 2 i}{2})}^{3} = x + i y \Rightarrow i^{3} - (- i)^{3} = x + i y \Rightarrow - i - i = x + i y \Rightarrow - 2 i = x + i y$

Comparing the real and imaginary parts,

(x, y) = (0, -2)

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Similar questions

Q. If

{(\frac{1 + i}{1 - i})}^{3} - {(\frac{1 - i}{1 + i})}^{3} = x + i y

, find (x, y).

Q. If

{(\frac{1 + i}{1 - i})}^{3} - {(\frac{1 - i}{1 + i})}^{3} = A + i B

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A, B =

Q. Find the real values of x and y, if
(i)

(x + i y) (2 - 3 i) = 4 + i

(ii)

(3 x - 2 i y) (2 + i)^{2} = 10 (1 + i)

(iii)

\frac{(1 + i) x - 2 i}{3 + i} + \frac{(2 - 3 i) y + i}{3 - i}

(iv)

(1 + i) (x + i y) = 2 - 5 i

i^{3} = \frac{1}{i} = \frac{1}{i} \times \frac{i}{i} = \frac{- i}{1} = - i

Q. If

\frac{{(1 + i)}^{2}}{2 - i} = x + i y

, find x + y.

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If (1+i1−i)3−(1−i1+i)3=x+iy, find (x,y)

$If {(\frac{1 + i}{1 - i})}^{3} - {(\frac{1 - i}{1 + i})}^{3} = x + i y, find (x,y)$