Find Real x and y if ( 𝑥 − 𝑖 𝑦 ) ( 3 + 5 𝑖 ) (x−iy)(3+5i) Is the Conjugate of a Complex Number | Query Point Official

Find Real x and y if `\((x - i y)(3 + 5i)\)` Is the Conjugate of a Complex Number

Problem Statement

Find the real numbers x and y such that:

`(x - i y)(3 + 5i)` is the complex conjugate of `-6 - 24i`.

Concept: Complex Conjugate

The complex conjugate of `a + bi` is `a - bi`. It reflects the imaginary part. Knowing this helps in solving equations involving complex products.

Step 1: Write the Conjugate

The conjugate of `-6 - 24i` is:

`-6 + 24i`

Step 2: Set Up the Equation

So we want:

`(x - i y)(3 + 5i) = -6 + 24i`

Step 3: Multiply Complex Numbers

Multiply the two binomials:

`(x - i y)(3 + 5i) = x·3 + x·5i - i y·3 - i y·5i`

Simplify terms:

• `= 3x + 5i x - 3i y - 5i^2 y`
• `= 3x + 5i x - 3i y + 5y` (because `i^2 = -1`)

Combine real and imaginary parts:

`= (3x + 5y) + i(5x - 3y)`

Step 4: Compare with Target

Match real and imaginary parts with `-6 + 24i`:

• Real part: `3x + 5y = -6`
• Imaginary part: `5x - 3y = 24`

Step 5: Solve the System of Equations

Now solve the linear system:

From the first equation:

`3x + 5y = -6`

From the second equation:

`5x - 3y = 24`

Multiply the first equation by 3:

`9x + 15y = -18`

Multiply the second by 5:

`25x - 15y = 120`

Add the two:

`(9x + 25x) + (15y - 15y) = -18 + 120`

`34x = 102`

`x = 3`

Substitute back to find y:

`3(3) + 5y = -6` → `9 + 5y = -6` → `5y = -15` → `y = −3`

Final Answer

✅ `x = 3` and `y = -3`

Explanation

The key idea is to express complex multiplication in terms of real and imaginary parts and then match them to the conjugate of the given complex number. This converts the problem into a system of linear equations, which we solve using elimination. (Standard algebraic method)

FAQ

What is a complex conjugate?

It is obtained by changing the sign of the imaginary component of a complex number. For example, the conjugate of `a + bi` is `a - bi`.

Why match real and imaginary parts?

Because two complex numbers are equal only if their real parts are equal and their imaginary parts are equal. This lets us form a system of equations.

Where else is this technique used?

You see this method in solving electrical engineering problems, signal processing, and when converting between rectangular and polar forms of complex numbers.

Related Topics

See Mathematics Notes & MCQs for more problems on complex numbers and analytical geometry.