Learn If X Iy3 U Iv Then Show That Ux Vy 4x2 Y2

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

**Solution:**

(x+iy)^{3} = x^{3} + (iy)^{3} + 3x (iy)(x + iy) [Using (a + b)³ formula]

= x^{3} - iy^{3} + 3ix^{2}y - 3xy^{2 }[because i² = -1]

= (x^{3} - 3xy^{2}) + i (3x^{2}y - y^{3})

= u + iv (Given)

Comparing the real and imaginary parts,

u = x^{3} - 3xy^{2} and v = 3x^{2}y - y^{3}.

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

LHS = u/x + v/y

= (x^{3} - 3xy^{2}) / x + (3x^{2}y - y^{3})/y

= x^{2} - 3y^{2} + 3x^{2} - y^{2}

= 4x^{2} - 4y^{2}

= 4 (x^{2} - y^{2})

= RHS

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

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)

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