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Question
If (x + iy)3 = y + vi then show that `(y/x + "v"/y)` = 4(x2 – y2)
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Solution
(x + yi)3 = y + vi
∴ x3 + 3x2yi + 3xy2i2 + y3i3 = y + vi
∴ x3 + 3x2yi + 3xy2 (–1) – y3i = y + vi ...[∵ i2 = – 1, i3 = – i]
∴ x3 + 3x2yi - 3xy2 – iy3 = y + vi
∴ (x3 – 3xy2) + (3x2y – y3)i = y + vi
Equating real and imaginary parts, we get
y = x3 – 3xy2 and v = 3x2y – y3
∴ `y/x` = x2 – 3y2 and `"v"/y` = 3x2 – y2
Consider,
L.H.S = `y/x + "v"/y`
= x2 – 3y2 + 3x2 – y2
= 4x2 – 4y2
= 4(x2 – y2)
= R.H.S.
Hence proved.
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