Solveeit Logo
Login

Question

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

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

Explanation

Solution

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

Formula used: ⇒i2=−1.......................(A) \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)(x - iy)(3 + 5i) as mentioned in the solution hint. Hence,
 =(x−iy)(3+5i) =3x+5xi−3yi−5yi2...............(1)\ = (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) = 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)(x - iy)(3 + 5i) is −6−24i - 6 - 24i hence, we have to find the inverse of −6−24i - 6 - 24i which is −6−24i‾=−6+24i\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= (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,
⇒(3x+5y)=−6............(4)\Rightarrow (3x + 5y) = - 6............(4)
⇒(5x−3y)=24.................(5)\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,

⇒5(3x+5y)=5×(−6) ⇒15x+25y=−30............(6) \Rightarrow 5(3x + 5y) = 5 \times ( - 6) \\\ \Rightarrow 15x + 25y = - 30............(6)

And,
 ⇒3(5x−3y)=3×24 ⇒15x−9y=72.................(7)\ \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,
 ⇒15x+25y−15x+9y=−30−72 ⇒34y=−102 ⇒y=−10234 ⇒y=−3\ \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,
 ⇒3x+5(−3)=−6 ⇒3x=−6+15 ⇒x=93 ⇒x=3\ \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)(x - iy)(3 + 5i) is the conjugate of −6−24i - 6 - 24i are x=3x = 3 and y=−3y = - 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 x‾\overline x
If the imaginary term i is multiplied with i or on squaring i means (i)2{(i)^2} we will obtain the real term as -1.