Question

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

Answer 1

(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.