If `a, b,c` are all positive and in HP, then the roots of `ax^2 +2bx +c=0` are

Options

A. real
B. imaginary
C. rational
D. equal

Answer

To solve the problem, we need to determine the nature of the roots of the quadratic equation \( ax^2 + 2bx + c = 0 \) given that \( a, b, c \) are all positive and in harmonic progression (HP).

### Step 1: Understanding Harmonic Progression
If \( a, b, c \) are in HP, it means that their reciprocals \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) are in arithmetic progression (AP). Therefore, we can express this relationship as:
\[ \frac{1}{b} = \frac{1}{2} \left( \frac{1}{a} + \frac{1}{c} \right) \]
This simplifies to:
\[ 2b = \frac{ac}{a + c} \]
or
\[ b = \frac{2ac}{a + c} \]

### Step 2: Finding the Discriminant
The nature of the roots of the quadratic equation \( ax^2 + 2bx + c = 0 \) is determined by the discriminant \( D \), which is given by:
\[ D = B^2 - 4AC \]
In our case, substituting \( B = 2b \):
\[ D = (2b)^2 - 4ac = 4b^2 - 4ac \]

### Step 3: Substitute the Value of \( b \)
Now we substitute the value of \( b \) from our earlier step:
\[ D = 4\left(\frac{2ac}{a + c}\right)^2 - 4ac \]
Calculating \( D \):
\[ D = 4 \cdot \frac{4a^2c^2}{(a + c)^2} - 4ac \]
\[ D = \frac{16a^2c^2}{(a + c)^2} - 4ac \]
To combine these terms, we need a common denominator:
\[ D = \frac{16a^2c^2 - 4ac(a + c)^2}{(a + c)^2} \]

### Step 4: Expand and Simplify
Expanding the second term:
\[ D = \frac{16a^2c^2 - 4ac(a^2 + 2ac + c^2)}{(a + c)^2} \]
This simplifies to:
\[ D = \frac{16a^2c^2 - 4a^3c - 8a^2c^2 - 4ac^3}{(a + c)^2} \]
\[ D = \frac{(16a^2c^2 - 8a^2c^2) - 4a^3c - 4ac^3}{(a + c)^2} \]
\[ D = \frac{8a^2c^2 - 4a^3c - 4ac^3}{(a + c)^2} \]

### Step 5: Factor Out Common Terms
Factoring out \( 4ac \):
\[ D = \frac{4ac(2ac - a^2 - c^2)}{(a + c)^2} \]

### Step 6: Analyzing the Sign of \( D \)
Since \( a, b, c \) are all positive, \( ac > 0 \). We need to analyze the term \( 2ac - a^2 - c^2 \):
- Since \( a, b, c \) are in HP, it implies a certain relationship that leads to \( 2ac < a^2 + c^2 \) (by the properties of HP and AM-GM inequality).
- Thus, \( 2ac - a^2 - c^2 < 0 \).

### Conclusion
Since \( D < 0 \), the roots of the quadratic equation \( ax^2 + 2bx + c = 0 \) are imaginary.

### Final Answer
The roots of the equation \( ax^2 + 2bx + c = 0 \) are **imaginary**.

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