Finding Real Component of Complex Number

[thunc14]

Here's the problem:
Which is the real component of the complex number described by the following:
If you multiply it by its conjugate, you get 41. If you square it, the sum of its components is the opposite of the square of 2 less than the imaginary component.
a)-5
b)-4
c)4
d)5
e) Answer is not there

I'm stuck. The first part says (a+bi)(a-bi)=41. I foiled that out to (a^2 + b^2)=41. When it says square 'it', I'm not sure how to determine if it's of the form (a+bi) or (a-bi), or how to figure out what a and b are. Pretty lost on this one.

 

[Harry_the_cat]

Let the complex number (ie IT) be a + bi.

If you square "it" ie (a+bi)^2, the sum of its components is the opposite of the square of 2 less than the imaginary component.

I don't understand what is meant by "is the opposite of"? Is that the actual wording in the question as given??

 

[thunc14]

Let the complex number (ie IT) be a + bi.

If you square "it" ie (a+bi)^2, the sum of its components is the opposite of the square of 2 less than the imaginary component.

I don't understand what is meant by "is the opposite of"? Is that the actual wording in the question as given??

I don't understand that either. I copied the problem down word for word and double checked. I'm so confused as to how to progress in the problem, and that 'opposite of' is another corkscrew of uncertainty.

 

[Dr.Peterson]

Here's the problem:
Which is the real component of the complex number described by the following:
If you multiply it by its conjugate, you get 41. If you square it, the sum of its components is the opposite of the square of 2 less than the imaginary component.
a)-5
b)-4
c)4
d)5
e) Answer is not there

I'm stuck. The first part says (a+bi)(a-bi)=41. I foiled that out to (a^2 + b^2)=41. When it says square 'it', I'm not sure how to determine if it's of the form (a+bi) or (a-bi), or how to figure out what a and b are. Pretty lost on this one.

The grammar of that entire sentence is all twisted up, with unclear antecedents and order of operations, apparently intended to confuse you, as no one would say "the opposite of the square of 2"; they'd say "-4". They seem to want you to think "square root of 2". And is it just "the square of 2" whose "opposite" is to be taken, or the whole phrase following?

If you square it, the sum of its [the original, or the square?] components is the opposite [negative?] of the square of 2 less than the imaginary component [of what?].​

One possible interpretation is that if the number is [MATH]a+bi[/MATH], its square is [MATH](a^2 - b^2) + (2ab)i[/MATH], so the sum of the components of the square is [MATH](a^2 - b^2) + (2ab)[/MATH]. This is to be equal to (maybe) [MATH]-((2ab) - 2^2) = 4 - 2ab[/MATH], or (maybe) [MATH](2ab) - (-2^2) = 4 + 2ab[/MATH]. This yields one of these equations:

• [MATH](a^2 - b^2) + (2ab) = 4 - 2ab[/MATH]
• [MATH](a^2 - b^2) + (2ab) = 4 + 2ab[/MATH]

So your second equation to solve simplifies to one of

• [MATH]a^2 - b^2 + 4ab = 4[/MATH]
• [MATH]a^2 - b^2 = 4[/MATH]

Combining either of these with your first equation, [MATH]a^2 + b^2 = 41[/MATH], and I don't get any of the offered solutions for a.