Question

if(x-iy)(3+5i) is conjugate of -6-24i then find the values of real numbers x and y.

Answer 1

Hope it helps..........
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Answer 2

$We \: know \:that , \\</p><p>[tex] \pink { If \: z = a + ib \:then \:it's \: conjugate }\\\pink {\overline z= a - ib }$

$Now, Conjugate \: of \: -6 - 24i \\ is \: (-6+24i)\: ---(1)$

$(x-iy)(3+5i) = -6 + 24i \: (given) \: --(2)$

$\implies -6+24i \\= 3x + 5ix - 3iy + 5y\\= (3x + 5y)+ (-3y+5x)i$

/* Compare real parts , we get */

$-6 = 3x + 5y \: --(3)$

/* Compare imaginary parts , we get */

$24 = 5x - 3y \: ---(4)$

/* Multiplying equation (3) by 3 and equation (4) by 5 , we get */

$-18 = 9x + 15y \: --(5)$

$120 = 25x - 15y \: --(6)$

/* Add Equations (5) and (6) ,we get */

$\implies 102 = 34x$

$\implies x = \frac{102}{34}$

$\implies x = 3$

/* Put x =3 in equation (5) , we get */

$-18 = 9\times 3 + 15y$

$\implies -18 - 27 = 15y$

$\implies 15y = - 45$

$\implies y = \frac{-45}{15}$

$\implies y = -3$

Therefore.,

$\green { The \: value\: x =3 \:and \:y = -3 }$

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