Question

If x+iy³ = u+iv , then show that u/x + v/y.= 4(x²-y²).

Pls Solve A challenge to the math aryabhatt

Answer 1

$\huge{\boxed{\red{Question. :-}}}$

⭐If x+iy³ = u+iv , then show

that u/x + v/y.= 4(x²-y²).

$\huge{\boxed{Answer. :-}}$

✳$\huge\frac{u}{x} + \frac{v}{y} = 4x {}^{2} - 4y {}^{2}$

$\huge{\boxed{Solution:-}}$

Given:- x+iy³ = u+iv ,

To Prove:- u/x + v/y.= 4(x²-

y²).

Proof:-

(x+iy)³ = u+iv

=>x³+i³y³+3x²iy+3xi²y²=u+iv

=>x³-iy³+3x²iy-3xy²= u+iv

=> x²-3xy²=u

=>x(x²-3y²)=u

=>x²-3y²=u/x

3x²y - y³ = v

=>y(3x²-y²)=v

=> 3x²- y² = v/y

$\huge\frac{u}{x} + \frac{v}{y} = 4x {}^{2} - 4y {}^{2}$

Hence proved. ✔✔

# 101% Spam free Answer . ✔✔

Answer 2

$\huge{\text{\underline{Solution:-}}}$

$\implies$(x + iy)³ = u + iv

$\implies$x³ + 3x²iy + 3 xi²y² + i³y³ = u + iv

$\implies$x³ + 3x²iy - 3xy² - iy³ = u + iv

$\implies$(x³ - 3xy²) + i (3x²y - y³) = u + iv

★ Therefore,

$\implies$x³ - 3xy² = u and 3x²y - y³ = v

★ Therefore,

$\implies$u / x + v / y

$\implies$x³ - 3xy² / x + 3x²y - y³ / y

$\implies$x² - 3y² + 3x² - y²

$\implies$4x² - 4y²

$\implies$4 (x² - y²)

Hence proved!

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