if x+iy=(a+ib)³ then show that x/a+y/b=4(a²-b²)
Answers
Answer:
Proved below.
Step-by-step explanation:
x + iy = (a + ib)³
= a³ + (ib)³ + 3a²(ib) + 3a(ib)²
[∵ (a + b)³ = a³ + b³ + 3a²b + 3ab²]
See we know that
i = √-1
i² = i*i = (√-1)² = -1
i³ = i²*i = -1*√-1 = -i
i⁴ = i²*i² = (-1)*(-1) = 1
⇒x + iy = a³ + (ib)³ + 3a²(ib) + 3a(ib)²
= a³ + (-i)*b³ + 3a²(ib) + 3a(-1)b²
= a³ - ib³ + i*3a²b - 3ab²
⇒x + iy = a³- 3ab² + i(3a²b - b³)
Compare the real and imaginary coefficients in both sides
⇒x = a³- 3ab² | ⇒y = 3a²b - b³
Divde with 'a' on both sides | Divide with 'b' on both sides
⇒x/a = a² - 3b² _____ (1) | ⇒y/b = 3a² - b² ______(2)
Add (1) and (2)
⇒x/a + y/b = a² - 3b² + 3a² - b²
= 4a² - 4b²
=4(a² - b²)
∴x/a + y/b = 4(a² - b²)
Hence proved.