If (x + iy)3 = y + vi then show that (yx+vy) = 4(x2 – y2) - Mathematics and Statistics

Sum

If (x + iy)3 = y + vi then show that (y/x + "v"/y) = 4(x2 – y2)

Solution

(x + yi)3 = y + vi

∴ x3 + 3x2yi + 3xy2i2 + y3i3 = y + vi

∴ x3 + 3x2yi + 3xy2 (–1) – y3i = y + vi  ...[∵ i2 = – 1, i3 = – 1]

∴ (x3 – 3xy2) + (3x2y – y3)i = y + vi

Equating real and imaginary parts, we get

y = x3 – 3xy2 and v = 3x2y – y

∴ y/x = x2 – 3y2 and "v"/y = 3x2 – y2

∴ y/x + "v"/y = x2 – 3y2 + 3x2 – y2 = 4x2 – 4y2

∴ y/x + "v"/y = 4(x2 – y2)

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