Question

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

Answer 1

$Let z=(x-iy)(3+5i)$

$z=3x+5xi-3yi-5yi^2=3x+5xi-3yi+5y=(3x+5y)+i(5x-3y)$

$∴\bar{z}=(3x+5y)-i(5x-3y)=-6-24i$

Equating real and imaginary parts, we obtain

3x+5y=-6….(i)

5x-3y=24……(ii)

Multiplying equation (i) by 3 and equation (ii) by 5 and then adding them, we obtain

$9x+15y=-18$

$\frac{25x-15y=120}{34x=102}$

$∴ x=\frac{102}{34}=3$

Putting the value of x in equation (i), we obtain

$3(3)+5y=-6$

$⇒5y=-6-9=-15$

$⇒y=-3$

Thus, the values of x and y are 3 and -3 respectively.