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What is the quotient of the complex number 4-3i divided by its conjugate?
ANSWERS:
A. 24/25 + 7/25i
B. 24/25 - 7/25i
C. 7/25- 24/25i
D. 7/25 + 25/25i
Need an answer ASAP

Respuesta :

carlosego

Answer: C. [tex]\frac{7}{25}-\frac{24}{25}i[/tex]


Step-by-step explanation:

1. You have the following division:

[tex]\frac{4-3i}{4+3i}[/tex] (As you can see, to find the conjugate of  4-3i you must change the sign between the terms).

2. To solve this division, you must multiply the numerator and the denominator by the conjugate of the denominator, as following:

[tex]=\frac{(4-3i)}{(4+3i)}\frac{(4-3i)}{(4-3i)}=\frac{16-12i-12i+9i^{2}}{16-9i^{2}}[/tex]

3. Keeping on mind that [tex]i^{2}=-1[/tex], you have:

[tex]=\frac{16-12i-12i+9(-1)}{16-9(-1)}[/tex]

4. Simplifying:

[tex]=\frac{7-24i}{25}=\frac{7}{25}-\frac{24}{25}i[/tex]

5. The result is:

[tex]\frac{7}{25}-\frac{24}{25}i[/tex]

snehashish65

The quotient of the complex number 4 - 3i divided by its conjugate is  [tex]\dfrac{7}{25} -\dfrac{24}{25}i[/tex]. And option (C) is correct.

Given data:

The complex number is, 4 - 3i.

To find: The quotient when divided by conjugate of given complex number.

For a given complex number say, a + bi, the conjugate is given as,

a - bi

Then, the conjugate of the given complex number is, 4 + 3i.

Divide the given complex number with its conjugate as,

[tex]=\dfrac{ 4 - 3i}{4 + 3i}\\\\=\dfrac{ 4 - 3i}{4 + 3i} \times \dfrac{4-3i}{4-3i} \\\\=\dfrac{(4-3i)^{2}}{4^{2}-(3i)^{2}}[/tex]

Since, [tex]i^{2} =-1[/tex].

Solving further as,

[tex]=\dfrac{4^{2}+(3i)^{2}-2(4)(3i)}{16-9(-1)} \\\\=\dfrac{16+9(-1)-24)}{25}} \\\\=\dfrac{7}{25} -\dfrac{24}{25}i[/tex]

Thus, we can conclude that the quotient of the complex number 4 - 3i divided by its conjugate is  [tex]\dfrac{7}{25} -\dfrac{24}{25}i[/tex]. And option (C) is correct.

Learn more about the conjugate of complex number here:

https://brainly.com/question/18392150

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