Question

Find the values of x and y which satisfy the following equations (x, y ∈ R):

`(x + 1)/(1 + "i") + (y - 1)/(1 - "i")` = i

Answer 1

`(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")/(1^2 - "i"^2)`

∴ `((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.