Question:

Step 1: Express the complex numbers in rectangular form and state the given conditions. Let the two non-zero complex numbers be $z_1 = x_1 + iy_1$ and $z_2 = x_2 + iy_2$, where $x_1, y_1, x_2, y_2$ are real numbers. Since $z_1$ and $z_2$ are non-zero, it implies that $(x_1, y_1) \ne (0,0)$ and $(x_2, y_2) \ne (0,0)$. The given conditions are: 1. $\text{Re}(z_1z_2) = 0$ 2. $\text{Re}(z_1+z_2) = 0$ Step 2: Simplify the second condition $\text{Re}(z_1+z_2)=0$. First, calculate $z_1+z_2$: $$z_1+z_2 = (x_1+iy_1) + (x_2+iy_2) = (x_1+x_2) + i(y_1+y_2)$$ The real part of $z_1+z_2$ is $x_1+x_2$. So, the condition $\text{Re}(z_1+z_2)=0$ implies: $$x_1+x_2 = 0 \implies x_2 = -x_1$$ Step 3: Simplify the first condition $\text{Re}(z_1z_2)=0$. Next, calculate $z_1z_2$: $$z_1z_2 = (x_1+iy_1)(x_2+iy_2) = x_1x_2 + ix_1y_2 + ix_2y_1 + i^2y_1y_2$$ $$z_1z_2 = (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1)$$ The real part of $z_1z_2$ is $x_1x_2 - y_1y_2$. So, the condition $\text{Re}(z_1z_2)=0$ implies: $$x_1x_2 - y_1y_2 = 0$$ Step 4: Combine the simplified conditions to find a relationship between $y_1$ and $y_2$. Substitute $x_2 = -x_1$ (from Step 2) into the equation from Step 3: $$x_1(-x_1) - y_1y_2 = 0$$ $$-x_1^2 - y_1y_2 = 0$$ This can be rewritten as: $$x_1^2 + y_1y_2 = 0$$ Step 5: Analyze the derived relationship for $y_1$ and $y_2$. From $x_1^2 + y_1y_2 = 0$, we have $y_1y_2 = -x_1^2$. Since $x_1$ is a real number, $x_1^2 \ge 0$. Also, $z_1$ and $z_2$ are non-zero. If $x_1=0$, then $y_1y_2 = -0^2 = 0$. If $x_1=0$, then from $x_2=-x_1$, we have $x_2=0$. So, $z_1=iy_1$ and $z_2=iy_2$. For $z_1$ and $z_2$ to be non-zero, $y_1 \ne 0$ and $y_2 \ne 0$. However, $y_1y_2=0$ with $y_1 \ne 0$ and $y_2 \ne 0$ is a contradiction. Therefore, our assumption $x_1=0$ must be false. This means $x_1 \ne 0$, which implies $x_1^2 > 0$. Consequently, $y_1y_2 = -x_1^2$ must be strictly negative: $$y_1y_2 < 0$$ This inequality implies that $y_1$ and $y_2$ must have opposite signs. That is, one must be positive and the other negative. Step 6: Evaluate the given options based on the derived condition. We found that $\text{Im}(z_1)$ and $\text{Im}(z_2)$ (which are $y_1$ and $y_2$ respectively) must have opposite signs. Let's check the options: (A) $\text{Im}(z_1)>0$ and $\text{Im}(z_2)>0 \implies y_1>0, y_2>0 \implies y_1y_2 > 0$. This contradicts $y_1y_2 < 0$. So, (A) is not possible. (B) $\text{Im}(z_1)<0$ and $\text{Im}(z_2)>0 \implies y_1<0, y_2>0 \implies y_1y_2 < 0$. This is consistent with our condition. So, (B) is possible. (C) $\text{Im}(z_1)>0$ and $\text{Im}(z_2)<0 \implies y_1>0, y_2<0 \implies y_1y_2 < 0$. This is consistent with our condition. So, (C) is possible. (D) $\text{Im}(z_1)<0$ and $\text{Im}(z_2)<0 \implies y_1<0, y_2<0 \implies y_1y_2 > 0$. This contradicts $y_1y_2 < 0$. So, (D) is not possible. Thus, options (B) and (C) are possible. The final answer is $\boxed{\text{B and C}}$.