If `|a|=|b|=1 and |a+b|=sqrt(3)`, then the value of `(3a-4b)(2b+5b)` is

Options

A. `-21`
B. `-(21)/(2)`
C. `21`
D. `(21)/(2)`

Explanation

To solve the problem, we start with the given conditions:

1. \( |a| = |b| = 1 \)
2. \( |a + b| = \sqrt{3} \)

We need to find the value of \( (3a - 4b)(2a + 5b) \).

### Step 1: Use the given magnitudes
Since \( |a| = 1 \) and \( |b| = 1 \), we can express \( a \) and \( b \) in terms of their angles in the complex plane. Let's assume:
- \( a = e^{i\theta} \)
- \( b = e^{i\phi} \)

### Step 2: Find \( |a + b| \)
Using the property of magnitudes:
\[ |a + b|^2 = |a|^2 + |b|^2 + 2 \text{Re}(a\overline{b}) = 1 + 1 + 2 \text{Re}(a\overline{b}) = 2 + 2 \text{Re}(a\overline{b}) \]
Given that \( |a + b| = \sqrt{3} \), we have:
\[ |a + b|^2 = 3 \]
Thus,
\[ 2 + 2 \text{Re}(a\overline{b}) = 3 \]
This simplifies to:
\[ 2 \text{Re}(a\overline{b}) = 1 \implies \text{Re}(a\overline{b}) = \frac{1}{2} \]

### Step 3: Calculate \( a\overline{b} \)
The expression \( a\overline{b} \) can be interpreted as:
\[ \text{Re}(a\overline{b}) = \cos(\theta - \phi) = \frac{1}{2} \]
This means:
\[ \theta - \phi = \frac{\pi}{3} \text{ or } \theta - \phi = -\frac{\pi}{3} \]

### Step 4: Calculate \( (3a - 4b)(2a + 5b) \)
Now we expand the expression:
\[ (3a - 4b)(2a + 5b) = 3a \cdot 2a + 3a \cdot 5b - 4b \cdot 2a - 4b \cdot 5b \]
Calculating each term:
1. \( 3a \cdot 2a = 6a^2 = 6 \) (since \( |a|^2 = 1 \))
2. \( 3a \cdot 5b = 15ab \)
3. \( -4b \cdot 2a = -8ab \)
4. \( -4b \cdot 5b = -20b^2 = -20 \) (since \( |b|^2 = 1 \))

Combining these:
\[ (3a - 4b)(2a + 5b) = 6 + 15ab - 8ab - 20 = 6 - 20 + 7ab = -14 + 7ab \]

### Step 5: Find \( ab \)
From our earlier calculations, we know:
\[ |ab| = |a||b| = 1 \]
We also found that \( \text{Re}(ab) = \frac{1}{2} \). Therefore, we can express \( ab \) as:
\[ ab = \frac{1}{2} + i\sqrt{1 - \left(\frac{1}{2}\right)^2} = \frac{1}{2} + i\frac{\sqrt{3}}{2} \]
Thus, \( ab = e^{i\frac{\pi}{3}} \) or \( ab = e^{-i\frac{\pi}{3}} \).

### Step 6: Substitute \( ab \) back into the expression
Since \( |ab| = 1 \), we can use \( ab = \frac{1}{2} + i\frac{\sqrt{3}}{2} \) to find:
\[ (3a - 4b)(2a + 5b) = -14 + 7 \cdot 1 = -14 + 7 = -7 \]

### Final Answer
Thus, the value of \( (3a - 4b)(2a + 5b) \) is \( -7 \).