If `abs(a)=1, abs(b)=2" and "abs(a-2b)=4" then "abs(a+3b)` is equal to

Options

A. 8
B. `sqrt(51/2)`
C. `sqrt(19/2)`
D. `sqrt(77/2)`

Answer

To solve the problem, we need to find the value of \( |a + 3b| \) given that \( |a| = 1 \), \( |b| = 2 \), and \( |a - 2b| = 4 \).

### Step-by-Step Solution:

1. **Given Information**:
- \( |a| = 1 \) implies \( |a|^2 = 1^2 = 1 \).
- \( |b| = 2 \) implies \( |b|^2 = 2^2 = 4 \).
- \( |a - 2b| = 4 \) implies \( |a - 2b|^2 = 4^2 = 16 \).

2. **Using the Formula for Magnitude**:
We know that \( |x|^2 = x \cdot x \). Therefore, we can express \( |a - 2b|^2 \) as:
\[ |a - 2b|^2 = (a - 2b) \cdot (a - 2b) = a \cdot a - 4a \cdot b + 4b \cdot b \]
Substituting the known values:
\[ 16 = |a|^2 - 4(a \cdot b) + 4|b|^2 \]
\[ 16 = 1 - 4(a \cdot b) + 4 \cdot 4 \]
\[ 16 = 1 - 4(a \cdot b) + 16 \]
Simplifying gives:
\[ 16 = 17 - 4(a \cdot b) \]
\[ 4(a \cdot b) = 1 \implies a \cdot b = \frac{1}{4} \]

3. **Finding \( |a + 3b|^2 \)**:
Now we need to calculate \( |a + 3b|^2 \):
\[ |a + 3b|^2 = (a + 3b) \cdot (a + 3b) = a \cdot a + 6a \cdot b + 9b \cdot b \]
Substituting the known values:
\[ |a + 3b|^2 = |a|^2 + 6(a \cdot b) + 9|b|^2 \]
\[ |a + 3b|^2 = 1 + 6 \cdot \frac{1}{4} + 9 \cdot 4 \]
\[ |a + 3b|^2 = 1 + \frac{6}{4} + 36 \]
\[ |a + 3b|^2 = 1 + \frac{3}{2} + 36 \]
Converting \( 1 \) to a fraction:
\[ |a + 3b|^2 = \frac{2}{2} + \frac{3}{2} + \frac{72}{2} = \frac{77}{2} \]

4. **Taking the Square Root**:
Now, we take the square root to find \( |a + 3b| \):
\[ |a + 3b| = \sqrt{\frac{77}{2}} = \frac{\sqrt{77}}{\sqrt{2}} = \frac{\sqrt{77}}{\sqrt{2}} \text{ or } \sqrt{\frac{77}{2}} \]

### Final Answer:
Thus, the value of \( |a + 3b| \) is \( \sqrt{\frac{77}{2}} \).

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