Question

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

Explanation:

Step1: Identify conjugate of $4-3i$

The conjugate of $4-3i$ is $4+3i$.

Step2: Set up division expression

$$\frac{4-3i}{4+3i}$$

Step3: Multiply by conjugate of denominator

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

Step4: Expand numerator and denominator

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

Step5: Substitute $i^2=-1$

Numerator: $16-24i+9(-1)=16-24i-9=7-24i$
Denominator: $16-9(-1)=16+9=25$

Step6: Write final form

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

Answer:

$\frac{7}{25} - \frac{24}{25}i$ (second option)