If a, b are non-zero vectors such that `|a+b|=|a-2b|`, then
Statement-I Least value of `a*b+(4)/(|b|^(2)+2)` is `2sqrt(2)-1`.
Statement-II The expression `a*b+(4)/(|b|^(2)+2)` is least when magnitude of b is `sqrt(2tan((pi)/(8)))`.

Options

A. Both Statement-I and Statement-II are correct and Statement-II is the correct explanation of Statement-I
B. Both Statement-I and Statement-II are correct but Statement-II is not the correct explanation of Statement-I
C. Statement-I is correct but Statement-II is incorrect
D. Statement-II is correct but Statement-I is incorrect

Answer

To solve the problem, we start with the condition given in the question:

### Step 1: Given Condition
We have the condition:
\[ |a + b| = |a - 2b| \]
Squaring both sides, we get:
\[ |a + b|^2 = |a - 2b|^2 \]

### Step 2: Expanding the Squares
Expanding both sides:
\[ (a + b) \cdot (a + b) = (a - 2b) \cdot (a - 2b) \]
This leads to:
\[ |a|^2 + 2a \cdot b + |b|^2 = |a|^2 - 4a \cdot b + 4|b|^2 \]

### Step 3: Rearranging the Equation
Rearranging the equation gives:
\[ 2a \cdot b + |b|^2 = -4a \cdot b + 4|b|^2 \]
Combining like terms results in:
\[ 6a \cdot b = 3|b|^2 \]

### Step 4: Solving for \( a \cdot b \)
From the equation above, we can express \( a \cdot b \):
\[ a \cdot b = \frac{|b|^2}{2} \]

### Step 5: Expression to Minimize
Now, we need to analyze the expression given in Statement I:
\[ E = a \cdot b + \frac{4}{|b|^2 + 2} \]
Substituting \( a \cdot b \):
\[ E = \frac{|b|^2}{2} + \frac{4}{|b|^2 + 2} \]

### Step 6: Letting \( x = |b|^2 \)
Let \( x = |b|^2 \). Then, the expression becomes:
\[ E(x) = \frac{x}{2} + \frac{4}{x + 2} \]

### Step 7: Finding the Minimum Value
To find the minimum value, we differentiate \( E(x) \):
\[ \frac{dE}{dx} = \frac{1}{2} - \frac{4}{(x + 2)^2} \]
Setting the derivative to zero for critical points:
\[ \frac{1}{2} - \frac{4}{(x + 2)^2} = 0 \]
This leads to:
\[ \frac{4}{(x + 2)^2} = \frac{1}{2} \]
Cross-multiplying gives:
\[ 8 = (x + 2)^2 \]
Taking the square root:
\[ x + 2 = 2\sqrt{2} \quad \Rightarrow \quad x = 2\sqrt{2} - 2 \]

### Step 8: Evaluating the Minimum Value
Substituting \( x = 2\sqrt{2} - 2 \) back into \( E(x) \):
\[ E(2\sqrt{2} - 2) = \frac{2\sqrt{2} - 2}{2} + \frac{4}{(2\sqrt{2} - 2) + 2} \]
This simplifies to:
\[ E(2\sqrt{2} - 2) = \sqrt{2} - 1 + \frac{4}{2\sqrt{2}} = \sqrt{2} - 1 + 2\sqrt{2} = 3\sqrt{2} - 1 \]

### Step 9: Conclusion
Thus, the least value of \( E \) is:
\[ 2\sqrt{2} - 1 \]
This confirms that Statement I is correct.

### Step 10: Verifying Statement II
To verify Statement II, we need to find the magnitude of \( b \):
\[ |b|^2 = 2\sqrt{2} - 2 \quad \Rightarrow \quad |b| = \sqrt{2\sqrt{2} - 2} \]
Using the identity \( \tan(\frac{\pi}{8}) = \frac{\sqrt{2} - 1}{2} \), we find that:
\[ |b| = \sqrt{2 \tan(\frac{\pi}{8})} \]
Thus, Statement II is also correct.

### Final Answer
Both statements are correct, and Statement II correctly explains Statement I.