If a and b are distinct positive real numbers such that `a, a_(1), a_(2), a_(3), a_(4), a_(5), b` are in A.P. , `a, b_(1), b_(2), b_(3), b_(4), b_(5), b` are in G.P. and `a, c_(1), c_(2), c_(3), c_(4), c_(5), b` are in H.P., then the roots of `a_(3)x^(2)+b_(3)x+c_(3)=0` are

Options

A. real and distinct
B. real and equal
C. imaginary
D. none of these

Explanation

To solve the problem step by step, we will analyze the conditions given for the sequences and derive the necessary relationships to find the roots of the quadratic equation \(a_3x^2 + b_3x + c_3 = 0\).

### Step 1: Identify the means
Given that \(a, a_1, a_2, a_3, a_4, a_5, b\) are in Arithmetic Progression (A.P.), we can express \(a_3\) as the arithmetic mean of \(a\) and \(b\):
\[ a_3 = \frac{a + b}{2} \]

### Step 2: Identify the geometric mean
Similarly, since \(a, b_1, b_2, b_3, b_4, b_5, b\) are in Geometric Progression (G.P.), we can express \(b_3\) as the geometric mean of \(a\) and \(b\):
\[ b_3 = \sqrt{ab} \]

### Step 3: Identify the harmonic mean
For the harmonic progression \(a, c_1, c_2, c_3, c_4, c_5, b\), we find \(c_3\) as the harmonic mean of \(a\) and \(b\):
\[ c_3 = \frac{2ab}{a + b} \]

### Step 4: Establish the relationship between means
There is a known relationship between the arithmetic mean, geometric mean, and harmonic mean:
\[ b_3^2 = a_3 \cdot c_3 \]
Substituting the expressions we found:
\[ (\sqrt{ab})^2 = \left(\frac{a + b}{2}\right) \left(\frac{2ab}{a + b}\right) \]
This simplifies to:
\[ ab = ab \]
This confirms the relationship holds true.

### Step 5: Calculate the discriminant of the quadratic equation
The discriminant \(D\) of the quadratic equation \(a_3x^2 + b_3x + c_3 = 0\) is given by:
\[ D = b_3^2 - 4a_3c_3 \]
Substituting the values we found:
\[ D = (\sqrt{ab})^2 - 4\left(\frac{a + b}{2}\right)\left(\frac{2ab}{a + b}\right) \]
This simplifies to:
\[ D = ab - 4 \cdot \frac{ab}{2} = ab - 2ab = -ab \]
Since \(a\) and \(b\) are distinct positive real numbers, \(ab > 0\). Therefore:
\[ D = -ab < 0 \]

### Step 6: Conclusion about the roots
Since the discriminant \(D\) is negative, the roots of the quadratic equation \(a_3x^2 + b_3x + c_3 = 0\) are imaginary.

### Final Answer
The roots of the quadratic equation are **imaginary**.

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