Question

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

Explanation:

Step1: Identify the conjugate

The conjugate of $4-3i$ is $4+3i$. The quotient is $\frac{4-3i}{4+3i}$.

Step2: Multiply by conjugate of denominator

Multiply numerator and denominator by $4-3i$ (conjugate of denominator):
$\frac{(4-3i)(4-3i)}{(4+3i)(4-3i)}$

Step3: Expand numerator and denominator

Numerator: $(4-3i)^2 = 4^2 - 2(4)(3i) + (3i)^2 = 16 -24i +9i^2 = 16-24i-9=7-24i$
Denominator: $4^2 - (3i)^2 = 16 -9i^2 =16+9=25$

Step4: Simplify the fraction

$\frac{7-24i}{25} = \frac{7}{25} - \frac{24}{25}i$

Answer:

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