To solve the problem, we need to analyze the relationships between the distinct positive real numbers \( a \) and \( b \) based on the given conditions of being in Arithmetic Progression (A.P.), Geometric Progression (G.P.), and Harmonic Progression (H.P.).
### Step-by-Step Solution:
1. **Understanding the Progressions**:
- Since \( a, a_1, a_2, a_3, a_4, a_5, b \) are in A.P., we can express \( a_n \) as:
\[
a_n = a + n \cdot d \quad \text{for } n = 1, 2, 3, 4, 5
\]
where \( d \) is the common difference. Thus, \( a_3 = a + 2d \) and \( b = a + 6d \).
- Since \( a, b_1, b_2, b_3, b_4, b_5, b \) are in G.P., we can express \( b_n \) as:
\[
b_n = a \cdot r^n \quad \text{for } n = 1, 2, 3, 4, 5
\]
where \( r \) is the common ratio. Thus, \( b_3 = a \cdot r^3 \) and \( b = a \cdot r^6 \).
- Since \( a, c_1, c_2, c_3, c_4, c_5, b \) are in H.P., we can express \( c_n \) as:
\[
c_n = \frac{2ab}{a + b} \quad \text{for } n = 1, 2, 3, 4, 5
\]
Thus, \( c_3 = \frac{2ab}{a + b} \).
2. **Finding the Means**:
- The arithmetic mean \( a_3 \) is given by:
\[
a_3 = \frac{a + b}{2} = \frac{a + (a + 6d)}{2} = a + 3d
\]
- The geometric mean \( b_3 \) is given by:
\[
b_3 = \sqrt{ab}
\]
- The harmonic mean \( c_3 \) is given by:
\[
c_3 = \frac{2ab}{a + b}
\]
3. **Using the Relationship**:
- We know that for any two numbers \( a \) and \( b \):
\[
b_3^2 = a_3 \cdot c_3
\]
- Substituting the means we found:
\[
(\sqrt{ab})^2 = (a + 3d) \cdot \left(\frac{2ab}{a + b}\right)
\]
- Simplifying gives:
\[
ab = (a + 3d) \cdot \left(\frac{2ab}{a + b}\right)
\]
4. **Discriminant Calculation**:
- The roots of the quadratic equation \( a_3 x^2 + b_3 x + c_3 = 0 \) have a discriminant \( D \) given by:
\[
D = b_3^2 - 4a_3c_3
\]
- Substituting the values:
\[
D = (\sqrt{ab})^2 - 4(a + 3d) \cdot \left(\frac{2ab}{a + b}\right)
\]
- Simplifying:
\[
D = ab - 8 \cdot \frac{(a + 3d)ab}{a + b}
\]
- Since \( D < 0 \), the roots are imaginary.
### Conclusion:
The roots of the quadratic equation \( a_3 x^2 + b_3 x + c_3 = 0 \) are **imaginary roots**.
