IDNLearn.com offers a seamless experience for finding and sharing knowledge. Our platform provides trustworthy answers to help you make informed decisions quickly and easily.

Which complex number has an absolute value of 5?

A. [tex] -3 + 4i [/tex]

B. [tex] 2 + 3i [/tex]

C. [tex] 7 - 2i [/tex]

D. [tex] 9 + 4i [/tex]


Sagot :

To find the complex number that has an absolute value of 5, let's analyze each of the provided options. The absolute value (or modulus) of a complex number [tex]\( a + bi \)[/tex] is given by the formula [tex]\( |a + bi| = \sqrt{a^2 + b^2} \)[/tex]. We will calculate the absolute value for each given complex number.

1. For [tex]\(-3 + 4i\)[/tex]:
[tex]\[ | -3 + 4i | = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \][/tex]

2. For [tex]\(2 + 3i\)[/tex]:
[tex]\[ | 2 + 3i | = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.605 \][/tex]

3. For [tex]\(7 - 2i\)[/tex]:
[tex]\[ | 7 - 2i | = \sqrt{7^2 + (-2)^2} = \sqrt{49 + 4} = \sqrt{53} \approx 7.280 \][/tex]

4. For [tex]\(9 + 4i\)[/tex]:
[tex]\[ | 9 + 4i | = \sqrt{9^2 + 4^2} = \sqrt{81 + 16} = \sqrt{97} \approx 9.849 \][/tex]

From the calculations above, we see that the complex number [tex]\(-3 + 4i\)[/tex] has an absolute value of 5. Therefore, the complex number with an absolute value of 5 is:
[tex]\[ \boxed{-3+4i} \][/tex]