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Question
Find the value of x and y which satisfy the following equation (x, y∈R).
`(x+ 1)/(1 + "i") + (y - 1)/(1 - "i")` = i
Solution
`(x+ 1)/(1 + "i") + (y - 1)/(1 - "i")` = i
∴ `((x + 1)(1 - "i") + (y - 1)(1 + "i"))/((1 + "i")(1 - "i"))` = i
∴ `(x - x"i" + 1 - "i" + y + y"i" - 1 - "i")/(1 - "i"^2)` = i
∴ `((x + y) + (y - x - 2)"i")/(1 - (-1))` = i ...[∵ i2 = – 1]
∴ (x + y) + (y – x – 2)i = 2i
∴ (x + y) + (y – x – 2)i = 0 + 2i
Equating real and imaginary parts, we get
x + y = 0 and y – x – 2 = 2
∴ x + y = 0 ...(i)
and – x + y = 4 ...(ii)
Adding (i) and (ii), we get
2y = 4
∴ y = 2
Putting y = 2 in (i), we get
x + 2 = 0
x = – 2
x = – 2 and y = 2
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