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A day full of math games & activities. Find one near you.
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)
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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