If the complex number $x^{2} + y^{2} + 100 i$ and $29 - x^{2} y^{2} i$ are conjugate to each other, Find the value of $x^{3} + y^{3}$ when x and y are positive real number.

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Solution

: Let $z_{1}$ = $x^{2} + y^{2} + 100 i$

$z_{2}$ = $29 - x^{2} y^{2} i$

Conjugate of $z_{1}$ = $x^{2} + y^{2} + 100 i$

Given

$x^{2} + y^{2} - 100 i$ = $29 - x^{2} y^{2} i$

Comparing real and imaginary part on both sides

$x^{2} + y^{2}$ = 29 - - - - - - - (1)

$x^{2} y^{2}$ = 100 - - - - - - - (2)

Substitute y2 = in equation 1

$x^{2}$ + $\frac{100}{x^{2}}$ = 29

Let $x^{2}$ = t

t + $\frac{100}{t}$ = 29

$t^{2} - 29 t + 100$ = 0

(t - 25) (t - 4) = 0

t = 25, t = 4

$x^{2}$ = 25, $x^{2}$ = 4

x = $\pm 5$ , x = $\pm 2$

x should be positive real number

x = 5, 2

Substitute x in equation 2

y = 2, 5

When x = 5, y = 2

x =2, y = 5

$x^{3} + y^{3}$ = $5^{3} + 2^{3}$ = 125 + 8 = 133

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