Let an angle between a and b be `2pi//3`. If `abs(b)=2abs(a)` and the vectors a + xb and a - b are at right angles, then the value of x is:

Options

A. `2//3`
B. `2//5`
C. `1//3`
D. `1//5`

Answer

To solve the problem step by step, we will use the properties of vectors and the dot product.

### Step 1: Understand the Given Information
We are given:
- The angle between vectors **a** and **b** is \( \frac{2\pi}{3} \).
- The magnitude of **b** is twice the magnitude of **a**: \( |b| = 2|a| \).
- The vectors **a + xb** and **a - b** are perpendicular.

### Step 2: Set Up the Dot Product Condition
Since the vectors **a + xb** and **a - b** are perpendicular, their dot product is zero:
\[ (a + xb) \cdot (a - b) = 0 \]

### Step 3: Expand the Dot Product
Expanding the left-hand side using the distributive property of the dot product:
\[ a \cdot a - a \cdot b + x(b \cdot a) - x(b \cdot b) = 0 \]
This simplifies to:
\[ |a|^2 - a \cdot b + x(a \cdot b) - x|b|^2 = 0 \]

### Step 4: Substitute Known Values
We know:
- \( |b| = 2|a| \), so \( |b|^2 = 4|a|^2 \).
- The dot product \( a \cdot b = |a||b| \cos\left(\frac{2\pi}{3}\right) \).
- Since \( \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \), we have:
\[ a \cdot b = |a| \cdot 2|a| \cdot \left(-\frac{1}{2}\right) = -|a|^2 \]

### Step 5: Substitute into the Equation
Substituting \( a \cdot b \) and \( |b|^2 \) into the equation:
\[ |a|^2 - (-|a|^2) + x(-|a|^2) - x(4|a|^2) = 0 \]
This simplifies to:
\[ |a|^2 + |a|^2 - x|a|^2 - 4x|a|^2 = 0 \]
\[ 2|a|^2 - x|a|^2(1 + 4) = 0 \]
\[ 2|a|^2 - 5x|a|^2 = 0 \]

### Step 6: Factor Out \( |a|^2 \)
Assuming \( |a|^2 \neq 0 \) (since **a** is a vector):
\[ 2 - 5x = 0 \]

### Step 7: Solve for \( x \)
Rearranging gives:
\[ 5x = 2 \implies x = \frac{2}{5} \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{\frac{2}{5}} \]