It is given that $a_{11}, a_{22}$ are real numbers, that $x_1, x_2, a_{12}, b_1, b_2$ are complex numbers, and that $a_{11}a_{22}=a_{12}\overline{a_{12}}$ (Where $\overline{a_{12}}$ is he conjugate of $a_{12}$). We consider the following system in $x_1, x_2$:
\[\overline{x_1}(a_{11}x_1 + a_{12}x_2) = b_1,\]\[\overline{x_2}(a_{12}x_1 + a_{22}x_2) = b_2.\]
[b](a) [/b]Give one condition to make the system consistent.

[b](b) [/b]Give one condition to make $\arg x_1 - \arg x_2 = 98^{\circ}.$
1. **Given Information and Equations:**
   - Real numbers: \(a_{11}, a_{22}\)
   - Complex numbers: \(x_1, x_2, a_{12}, b_1, b_2\)
   - Condition: \(a_{11}a_{22} = a_{12}\overline{a_{12}} = |a_{12}|^2\)
   - System of equations:
     \[
     \overline{x_1}(a_{11}x_1 + a_{12}x_2) = b_1,
     \]
     \[
     \overline{x_2}(a_{12}x_1 + a_{22}x_2) = b_2.
     \]

2. **Rewrite the Equations:**
   - Expanding the given equations:
     \[
     a_{11}|x_1|^2 + a_{12}\overline{x_1}x_2 = b_1,
     \]
     \[
     a_{22}|x_2|^2 + a_{12}\overline{x_2}x_1 = b_2.
     \]
   - Taking the conjugate of these equations:
     \[
     a_{11}|x_1|^2 + \overline{a_{12}} x_1 \overline{x_2} = \overline{b_1},
     \]
     \[
     a_{22}|x_2|^2 + \overline{a_{12}} x_2 \overline{x_1} = \overline{b_2}.
     \]

3. **Eliminate Cross Terms:**
   - Subtract the conjugate equations from the original equations to eliminate the cross terms:
     \[
     a_{11}|x_1|^2 + a_{12}\overline{x_1}x_2 - (a_{11}|x_1|^2 + \overline{a_{12}} x_1 \overline{x_2}) = b_1 - \overline{b_1},
     \]
     \[
     a_{22}|x_2|^2 + a_{12}\overline{x_2}x_1 - (a_{22}|x_2|^2 + \overline{a_{12}} x_2 \overline{x_1}) = b_2 - \overline{b_2}.
     \]
   - Simplifying these equations:
     \[
     a_{12}\overline{x_1}x_2 - \overline{a_{12}} x_1 \overline{x_2} = b_1 - \overline{b_1},
     \]
     \[
     a_{12}\overline{x_2}x_1 - \overline{a_{12}} x_2 \overline{x_1} = b_2 - \overline{b_2}.
     \]

4. **Solve for \(|x_1|^2\) and \(|x_2|^2\):**
   - Using the given condition \(a_{11}a_{22} = |a_{12}|^2\), we can solve for \(|x_1|^2\) and \(|x_2|^2\):
     \[
     |x_1|^2 = \frac{1}{a_{11}(a_{12}^2 - \overline{a_{12}}^2)} \left[(\overline{b_2} - b_2)a_{11}a_{22} + (\overline{b_1}a_{12}^2 - b_1\overline{a_{12}}^2)\right],
     \]
     \[
     |x_2|^2 = \frac{1}{a_{22}(a_{12}^2 - \overline{a_{12}}^2)} \left[(\overline{b_1} - b_1)a_{11}a_{22} + (\overline{b_2}a_{12}^2 - b_2\overline{a_{12}}^2)\right].
     \]

5. **Consistency Condition:**
   - Rewrite the first of the original equations as:
     \[
     a_{12}\overline{x_1}x_2 = b_1 - a_{11}|x_1|^2,
     \]
     \[
     \overline{a_{12}}x_1\overline{x_2} = \overline{b_1} - a_{11}|x_1|^2.
     \]
   - Multiply and add the equations:
     \[
     (a_{11}|x_1|^2 - a_{22}|x_2|^2)^2 + |b_1|^2 + |b_2|^2 - [a_{11}(b_1 + \overline{b_1})|x_1|^2 + a_{22}(b_2 + \overline{b_2})|x_2|^2] = 0.
     \]
   - Substituting the values of \(|x_1|^2\) and \(|x_2|^2\) in the above expression, we get the requested condition.

6. **Condition for \(\arg x_1 - \arg x_2 = 98^\circ\):**
   - Let \(\arg(x_1) = \theta_1\) and \(\arg(x_2) = \theta_2\), so that \(\tan \theta_1 = \frac{\text{Im}(x_1)}{\text{Re}(x_1)}\) and \(\tan \theta_2 = \frac{\text{Im}(x_2)}{\text{Re}(x_2)}\).
   - From the original equations, considering their imaginary parts, we get:
     \[
     \tan(\arg a_{12}) = \tan(98^\circ) \cdot \frac{\text{Im}(b_2 + b_1)}{\text{Im}(b_2 - b_1)}.
     \]

The final answer is \(\boxed{\tan(\arg a_{12}) = \tan(98^\circ) \cdot \frac{\text{Im}(b_2 + b_1)}{\text{Im}(b_2 - b_1)}}\)