Question

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

Answer 1

Hint: According to the question given in the question we have to find the real numbers x and y if $(x - iy)(3 + 5i)$ is the conjugate of $ - 6 - 24i$. So, first of all we have to determine the multiplication of the terms of the expression $(x - iy)(3 + 5i)$.
Now, we have to separate the real and imaginary terms of the expression obtained and as given that the conjugate of $(x - iy)(3 + 5i)$ is $ - 6 - 24i$ so we have to obtained the inverse of the $ - 6 - 24i$ by which we can obtained the values x and y.

Formula used: $ \Rightarrow {i^2} = - 1.......................(A)$

Complete step-by-step solution:
Step 1: First of all we have to multiply the terms of the expression $(x - iy)(3 + 5i)$ as mentioned in the solution hint. Hence,
$\
   = (x - iy)(3 + 5i) \\
   = 3x + 5xi - 3yi - 5y{i^2}...............(1)
 $
Step 2: Now, to solve the expression we have to use the identity (1) as mentioned in the solution hint.
\[\
   = 3x + 5xi - 3yi - 5y( - 1) \\
   = 3x + 5xi - 3yi + 5y \\
   = (3x + 5y) + i(5x - 3y)...........(2)
 \]
Step 3: Now, as we know that the conjugate of $(x - iy)(3 + 5i)$ is $ - 6 - 24i$ hence, we have to find the inverse of $ - 6 - 24i$ which is $\overline { - 6 - 24i} = - 6 + 24i$ so, on substituting the inverse obtained in the expression (2) as obtained in the solution step 2.
\[ = (3x + 5y) + i(5x - 3y) = - 6 + 24i\]……………..(3)
Step 4: Now, to obtain the values of x and y we have to compare the real and imaginary terms of the expression (3) as obtained in the solution step 3. Hence,
\[ \Rightarrow (3x + 5y) = - 6............(4)\]
\[ \Rightarrow (5x - 3y) = 24.................(5)\]
Step 5: Now, we have to solve the obtained expressions (4) and (5) but before that we have to multiply the equation (4) with 5 and equation (5) with 3. Hence, obtained equations are,
\[\
   \Rightarrow 5(3x + 5y) = 5 \times ( - 6) \\
   \Rightarrow 15x + 25y = - 30............(6)
 \]
And,
$\
   \Rightarrow 3(5x - 3y) = 3 \times 24 \\
   \Rightarrow 15x - 9y = 72.................(7)
 $
Step 6: Now, we have to subtract the expression (7) from the expression (6). Hence,
$\
   \Rightarrow 15x + 25y - 15x + 9y = - 30 - 72 \\
   \Rightarrow 34y = - 102 \\
   \Rightarrow y = - \dfrac{{102}}{{34}} \\
   \Rightarrow y = - 3
 $
Step 7: Now, to obtain the value of x we have to substitute the value of y in equation (4). Hence,
$\
   \Rightarrow 3x + 5( - 3) = - 6 \\
   \Rightarrow 3x = - 6 + 15 \\
   \Rightarrow x = \dfrac{9}{3} \\
   \Rightarrow x = 3
 $

Hence, with the help of identity (A) as mentioned in the solution hint we have obtained the values of x and y for if $(x - iy)(3 + 5i)$ is the conjugate of $ - 6 - 24i$ are $x = 3$ and $y = - 3$

Note: If the conjugate for any complex equation/expression is given then to solve the given equation/expression it is necessary to find the inverse of that given conjugate as if the conjugate of the given equation/expression is x then the inverse of that given conjugate is $\overline x $
If the imaginary term i is multiplied with i or on squaring i means ${(i)^2}$ we will obtain the real term as -1.