If (x + iy)³ = u + iv, then show that u/x + v/y = 4(x² - y²)

Solution:

(x+iy)³ = x³ + (iy)³ + 3x (iy)(x + iy) [Using (a + b)³ formula]

= x³ - iy³ + 3ix²y - 3xy²  [because i² = -1]

= (x³ - 3xy²) + i (3x²y - y³)

= u + iv (Given)

Comparing the real and imaginary parts,

u = x³ - 3xy² and v = 3x²y - y³.

Now, we will consider the LHS of what needs to be proved.

LHS = u/x + v/y

= (x³ - 3xy²) / x + (3x²y - y³)/y

= x² - 3y² + 3x² - y²

= 4x² - 4y²

= 4 (x² - y²)

= RHS

Hence we proved that u/x + v/y = 4(x² - y²)

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NCERT Solutions Class 11 Maths Chapter 5 Exercise ME Question 16

If (x + iy)³ = u + iv, then show that u/x + v/y = 4(x² - y²)

Summary:

If (x + iy)³ = u + iv, then we have shown that u/x + v/y = 4(x² - y²)