If $$ A(2a + 3b, a + 6b) $$ and $$ B(3a + 4b, 2a + 5b) $$ are two points, then the distance between the points $$ A $$ and $$ B $$ is:

Question

Accepted Answer

1. **Apply the Distance Formula**:  
   The formula for the distance between two points $$ A(x_1, y_1) $$ and $$ B(x_2, y_2) $$ is:  
   $$
   	ext{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.
   $$

Substituting the given coordinates of points:  
   - Point A: $$ (x_1, y_1) = (2a + 3b, a + 6b) $$.  
   - Point B: $$ (x_2, y_2) = (3a + 4b, 2a + 5b) $$.

Compute the differences:  
   $$
   x_2 - x_1 = (3a + 4b) - (2a + 3b) = a + b.
   $$
   $$
   y_2 - y_1 = (2a + 5b) - (a + 6b) = a - b.
   $$

2. **Substitute into the Formula**:  
   $$
   	ext{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.
   $$
   Substituting the values:  
   $$
   	ext{Distance} = \sqrt{(a + b)^2 + (a - b)^2}.
   $$

3. **Simplify the Expression**:  
   Expand both terms:  
   $$
   (a + b)^2 = a^2 + 2ab + b^2,
   $$
   $$
   (a - b)^2 = a^2 - 2ab + b^2.
   $$
   Add them together:  
   $$
   	ext{Distance} = \sqrt{(a^2 + 2ab + b^2) + (a^2 - 2ab + b^2)}.
   $$
   Simplify:  
   $$
   	ext{Distance} = \sqrt{2a^2 + 2b^2}.
   $$
   Factor out the common term:  
   $$
   	ext{Distance} = \sqrt{2(a^2 + b^2)}.
   $$

Thus, the distance between points A and B is **$$ \sqrt{2(a^2 + b^2)} $$**.

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### Final Answer:
The correct option is **D. $$ \sqrt{2(a^2 + b^2)} $$**.

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