Let the harmonic mean of two positive real numbers a and b be 4, If q is a positive real number such that a, 5, q, b is an arithmetic progression, then the value(s) of |q -a| is (are)

Options

A. 3,4
B. 2,5
C. 3,6
D. 6,9

Explanation

To solve the problem step by step, we will follow the given information about the harmonic mean and the arithmetic progression.

### Step 1: Use the definition of harmonic mean
The harmonic mean \( H \) of two positive real numbers \( a \) and \( b \) is given by the formula:
\[ H = \frac{2ab}{a + b} \]
We are given that the harmonic mean is 4, so we set up the equation:
\[ \frac{2ab}{a + b} = 4 \]
Multiplying both sides by \( a + b \) gives:
\[ 2ab = 4(a + b) \]
This simplifies to:
\[ 2ab = 4a + 4b \]
Rearranging gives us:
\[ 2ab - 4a - 4b = 0 \]
Dividing the entire equation by 2:
\[ ab - 2a - 2b = 0 \quad \text{(Equation 1)} \]

### Step 2: Set up the arithmetic progression
We know that \( a, 5, q, b \) is an arithmetic progression. In an arithmetic progression, the second term can be expressed as:
\[ 5 = a + d \]
where \( d \) is the common difference. Thus, we can express \( a \) in terms of \( d \):
\[ a = 5 - d \]
The fourth term \( b \) can be expressed as:
\[ b = a + 3d = (5 - d) + 3d = 5 + 2d \quad \text{(Equation 2)} \]

### Step 3: Substitute \( a \) and \( b \) into Equation 1
Now we substitute \( a \) and \( b \) from Equations 1 and 2 into the first equation:
\[ (5 - d)(5 + 2d) - 2(5 - d) - 2(5 + 2d) = 0 \]
Expanding this gives:
\[ 25 + 10d - 5d - 2d^2 - 10 + 2d - 10 - 4d = 0 \]
Combining like terms:
\[ 25 - 10 - 10 + 10d - 5d - 4d - 2d^2 = 0 \]
Simplifying further:
\[ 5 - 2d^2 + d = 0 \]
Rearranging gives:
\[ 2d^2 - 3d - 5 = 0 \quad \text{(Equation 3)} \]

### Step 4: Solve the quadratic equation
Now we will solve the quadratic equation \( 2d^2 - 3d - 5 = 0 \) using the quadratic formula:
\[ d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 2 \), \( b = -3 \), and \( c = -5 \):
\[ d = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-5)}}{2 \cdot 2} \]
Calculating the discriminant:
\[ d = \frac{3 \pm \sqrt{9 + 40}}{4} = \frac{3 \pm \sqrt{49}}{4} = \frac{3 \pm 7}{4} \]
This gives us two possible values for \( d \):
\[ d = \frac{10}{4} = \frac{5}{2} \quad \text{and} \quad d = \frac{-4}{4} = -1 \]

### Step 5: Calculate \( |q - a| \)
Recall that \( q \) is the third term in the arithmetic progression:
\[ q = a + 2d \]
Thus,
\[ q - a = 2d \]
Now we calculate \( |q - a| \):
1. For \( d = \frac{5}{2} \):
\[ |q - a| = |2 \cdot \frac{5}{2}| = |5| = 5 \]
2. For \( d = -1 \):
\[ |q - a| = |2 \cdot (-1)| = |-2| = 2 \]

### Final Answer
The values of \( |q - a| \) are \( 2 \) and \( 5 \).