Answer:
[tex]\displaystyle pm - qn = 0\text{ and } pn + qm \neq 0[/tex]
Or:
[tex]pm = qn \text{ and } pn + qm \neq 0[/tex]
Step-by-step explanation:
We are given two complex numbers:
[tex]\displaystyle m + ni \text{ and } p + qi[/tex]
Where m, n, p, and q are real numbers.
And we want to determine the relationship among m, n, p, and q such that the product of the two complex numbers will only have an imaginary part.
Find the product:
[tex]\displaystyle \begin{aligned} (m+ni)(p+qi) &= p(m+ni) + qi(m+ni) \\ &= (pm + pni) + (qmi + qni^2) \\ &= (pm + pni) + (qmi - qn) \\ &= (pm - qn) + i(pn + qm) \end{aligned}[/tex]
Therefore, for the product to have only an imaginary part, the real part must be zero and the imaginary part must not be zero. That is:
[tex]\displaystyle pm - qn = 0\text{ and } pn + qm \neq 0[/tex]
In conclusion, the relationship between m, n, p, and q is:
[tex]\displaystyle pm - qn = 0\text{ and } pn + qm \neq 0[/tex]
Or:
[tex]pm = qn \text{ and } pn + qm \neq 0[/tex]
(Note: I couldn't understand the provided answer choices, but the above answer is indeed correct (and the second equation can be ignored).)
