Question

Find real numbers x and y if (x-iy) (3-5i) is the conjugate of - 6-24i?
Keep signs in mind please! ​

Answer 1

Answer:

According to the question

x=3; y=-3

Answer 2

☆given

• ( x - iy) ( 3 + 5i) is the conjugate of -6 -24i

☆to find

• find the values of real number x and y

☆ formula

• [ i² = -1 ]

☆solution

The conjugate of -6 + 24i.

given, ( x - iy) ( 3 + 5i) = -6 + 24i

$\longmapsto\pmb \: x - iy = \dfrac{ - 6 + 24i}{3 + 5i} \\ \\ \\\longmapsto\pmb \: x - iy = \dfrac{( - 6 + 24i)(3 - 5i)}{(3 + 5i)(3 - 5i)} \\ \\ \\ \longmapsto\pmb \: x - iy = \frac{ - 18 + 30i + 72i - 120i^{2} }{ {(3)}^{2} - {(5i)}^{2} }$

$\longmapsto\pmb \: x - iy = \dfrac{ - 18 + 102i + 120}{9 - 25i^{2} } \qquad \qquad \red{ \{ {i}^{2} = - 1 \} }$

$\longmapsto\pmb \: x - iy = \dfrac{102 + 102i}{9 + 25} \\ \\ \\ \longmapsto\pmb \: x - iy = \frac{102 + 120i}{34} \\ \\ \\ \longmapsto\pmb \: x - iy = \frac{102}{34} + \frac{102i}{34} \\ \\ \\ \longmapsto\pmb \: x - iy = 3 + 3i$

$\qquad \qquad \bigg \{ \boxed{\therefore \: x = 3 \: and \: y = - 3} \bigg \}$

hence the value of x and y is 3 and -3