Question

what is the quotient of the complex number $4 - 3i$ divided by its conjugate?\
\\(\frac{24}{25} - \frac{7}{25}i\\) \\(\frac{7}{25} + \frac{24}{25}i\\) \\(\frac{7}{25} - \frac{24}{25}i\\) \\(\frac{24}{25} + \frac{7}{25}i\\)

Explanation:

Step1: Find the conjugate of \(4 - 3i\)

The conjugate of a complex number \(a - bi\) is \(a + bi\). So, the conjugate of \(4 - 3i\) is \(4 + 3i\).

Step2: Set up the division

We need to find \(\frac{4 - 3i}{4 + 3i}\). To simplify this, we multiply the numerator and denominator by the conjugate of the denominator, which is \(4 - 3i\).

Step3: Multiply the numerator and denominator

\[

$$\begin{align*} \frac{(4 - 3i)(4 - 3i)}{(4 + 3i)(4 - 3i)}&=\frac{16 - 12i - 12i + 9i^2}{16 - (3i)^2}\\ &=\frac{16 - 24i + 9(-1)}{16 - 9(-1)} \quad (\text{since } i^2 = -1)\\ &=\frac{16 - 24i - 9}{16 + 9}\\ &=\frac{7 - 24i}{25}\\ &=\frac{7}{25} - \frac{24}{25}i \end{align*}$$

\]

Answer:

\(\frac{7}{25} - \frac{24}{25}i\) (corresponding to the third option)