To determine the quotient of the complex number [tex]\( 4 - 3i \)[/tex] divided by its conjugate, we can follow these steps:
1. Identify the conjugate of the complex number:
The conjugate of [tex]\( 4 - 3i \)[/tex] is [tex]\( 4 + 3i \)[/tex].
2. Set up the division:
We need to compute:
[tex]\[
\frac{4 - 3i}{4 + 3i}
\][/tex]
3. Rationalize the denominator:
To simplify this expression, we multiply the numerator and the denominator by the conjugate of the denominator:
[tex]\[
\frac{4 - 3i}{4 + 3i} \times \frac{4 - 3i}{4 - 3i}
\][/tex]
4. Multiply the numerators and denominators:
The numerator becomes:
[tex]\[
(4 - 3i)(4 - 3i) = 16 - 12i - 12i + 9i^2 = 16 - 24i + 9(-1) = 16 - 24i - 9 = 7 - 24i
\][/tex]
The denominator becomes:
[tex]\[
(4 + 3i)(4 - 3i) = 16 - 9i^2 = 16 - 9(-1) = 16 + 9 = 25
\][/tex]
5. Rewrite and simplify the expression:
Combining the results from the numerator and the denominator, we get:
[tex]\[
\frac{4 - 3i}{4 + 3i} = \frac{7 - 24i}{25} = \frac{7}{25} - \frac{24}{25}i
\][/tex]
6. Final result:
The quotient of [tex]\( 4 - 3i \)[/tex] divided by its conjugate is:
[tex]\[
\frac{7}{25} - \frac{24}{25}i
\][/tex]
This matches the third option in the list given:
[tex]\[
\boxed{\frac{7}{25} - \frac{24}{25}i}
\][/tex]
