If one root of the quadratic equation `(a-b)x^2+ax+1=0` is double the other root where `a in R`, then the greatest value of b is

Options

A. `9//8`
B. `7//8`
C. `8//9`
D. `8//7`

Answer

To solve the problem, we need to find the greatest value of \( b \) in the quadratic equation \( (a-b)x^2 + ax + 1 = 0 \) given that one root is double the other root. Let's denote the roots as \( \alpha \) and \( 2\alpha \).

### Step 1: Set up the equations for the roots
From Vieta's formulas, we know:
1. The sum of the roots \( \alpha + 2\alpha = 3\alpha \) is equal to \( -\frac{a}{a-b} \).
2. The product of the roots \( \alpha \cdot 2\alpha = 2\alpha^2 \) is equal to \( \frac{1}{a-b} \).

### Step 2: Write the equations
From the sum of the roots:
\[ 3\alpha = -\frac{a}{a-b} \quad \text{(1)} \]
From the product of the roots:
\[ 2\alpha^2 = \frac{1}{a-b} \quad \text{(2)} \]

### Step 3: Express \( \alpha \) in terms of \( a \) and \( b \)
From equation (1):
\[ \alpha = -\frac{a}{3(a-b)} \quad \text{(3)} \]

### Step 4: Substitute \( \alpha \) into equation (2)
Substituting \( \alpha \) from equation (3) into equation (2):
\[ 2\left(-\frac{a}{3(a-b)}\right)^2 = \frac{1}{a-b} \]
This simplifies to:
\[ 2 \cdot \frac{a^2}{9(a-b)^2} = \frac{1}{a-b} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 2a^2 = 9(a-b) \quad \text{(4)} \]

### Step 6: Rearrange equation (4)
Rearranging equation (4) gives:
\[ 2a^2 = 9a - 9b \]
\[ 9b = 9a - 2a^2 \]
\[ b = a - \frac{2a^2}{9} \quad \text{(5)} \]

### Step 7: Find the maximum value of \( b \)
To find the maximum value of \( b \), we differentiate \( b \) with respect to \( a \):
\[ \frac{db}{da} = 1 - \frac{4a}{9} \]
Setting this equal to zero for maximization:
\[ 1 - \frac{4a}{9} = 0 \implies a = \frac{9}{4} \]

### Step 8: Substitute \( a \) back into equation (5)
Substituting \( a = \frac{9}{4} \) into equation (5):
\[ b = \frac{9}{4} - \frac{2\left(\frac{9}{4}\right)^2}{9} \]
Calculating \( \left(\frac{9}{4}\right)^2 = \frac{81}{16} \):
\[ b = \frac{9}{4} - \frac{2 \cdot \frac{81}{16}}{9} = \frac{9}{4} - \frac{162}{144} = \frac{9}{4} - \frac{9}{8} \]
Finding a common denominator (8):
\[ b = \frac{18}{8} - \frac{9}{8} = \frac{9}{8} \]

### Final Answer
Thus, the greatest value of \( b \) is:
\[ \boxed{\frac{9}{8}} \]