Question

the sum of two complex numbers, where the real numbers do not equal zero, results in a sum of 34i. which statement must be true about the complex numbers?
the complex numbers have equal imaginary parts.
the complex numbers have equal real parts.
the complex numbers have opposite imaginary parts.
the complex numbers have opposite real parts.

Explanation:

Step1: Define complex numbers

Let the two complex numbers be $z_1 = a + bi$ and $z_2 = c + di$, where $a
eq 0$, $c
eq 0$, and $a,b,c,d$ are real numbers.

Step2: Compute their sum

The sum is $z_1 + z_2 = (a + c) + (b + d)i$.

Step3: Match to given sum

We know the sum is $34i = 0 + 34i$. Equate real and imaginary parts:
Real part: $a + c = 0$
Imaginary part: $b + d = 34$

Step4: Analyze the results

From $a + c = 0$, we get $c = -a$, meaning the real parts are opposites. The imaginary parts sum to 34, so they are not equal or opposites.

Answer:

The complex numbers have opposite real parts.