If {(x + iy)^{1/3}} = a + ib, then frac{x}{a} + frac{y}{b} is equal to

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4-1.Complex numbers

medium

If ${(x + iy)^{1/3}} = a + ib,$then $\frac{x}{a} + \frac{y}{b}$is equal to

$4({a^2} + {b^2})$

$4({a^2} - {b^2})$

$4({b^2} - {a^2})$

None of these

Solution

(b) ${(x + iy)^{1/3}} = a + ib$==>$(x + iy) = {(a + ib)^3}$
$ = {a^3} + 3{a^2}.ib + 3a.{(ib)^2} + {(ib)^3}$
$ = {a^3} – 3a{b^2} + i(3{a^2}b – {b^3})$
Equating real and imaginary parts, we get
$\frac{x}{a} = {a^2} – 3{b^2}$and $\frac{y}{b} = 3{a^2} – {b^2}$
$\therefore $ $\frac{x}{a} + \frac{y}{b} = 4({a^2} – {b^2})$

Standard 11

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