ABCD is a parallelogram. L is a point on BC which divides BC in the ratio `1:2`. AL intersects BD at P.M is a point on DC which divides DC in the ratio `1 : 2` and AM intersects BD in Q.
`PQ : DB` is equal to

Options

A. `2//3`
B. `1//3`
C. `1//2`
D. `3//4`

Explanation

To solve the problem, we need to find the ratio \( PQ : DB \) in the given parallelogram \( ABCD \) with points \( L \) and \( M \) dividing segments \( BC \) and \( DC \) respectively.

### Step-by-Step Solution:

1. **Define the Vectors:**
Let \( \vec{A} = \vec{A} \) (point A), \( \vec{B} = \vec{A} + \vec{a} \) (point B), \( \vec{C} = \vec{B} + \vec{b} = \vec{A} + \vec{a} + \vec{b} \) (point C), and \( \vec{D} = \vec{A} + \vec{b} \) (point D).

2. **Find the Position of Point L:**
Point \( L \) divides \( BC \) in the ratio \( 1:2 \). Using the section formula:
\[ \vec{L} = \frac{2\vec{B} + 1\vec{C}}{2 + 1} = \frac{2(\vec{A} + \vec{a}) + 1(\vec{A} + \vec{a} + \vec{b})}{3} = \frac{3\vec{A} + 3\vec{a} + \vec{b}}{3} = \vec{A} + \vec{a} + \frac{1}{3}\vec{b} \]

3. **Find the Position of Point M:**
Point \( M \) divides \( DC \) in the ratio \( 1:2 \):
\[ \vec{M} = \frac{2\vec{D} + 1\vec{C}}{2 + 1} = \frac{2(\vec{A} + \vec{b}) + 1(\vec{A} + \vec{a} + \vec{b})}{3} = \frac{3\vec{A} + 2\vec{b} + \vec{a}}{3} = \vec{A} + \frac{1}{3}\vec{a} + \frac{2}{3}\vec{b} \]

4. **Find the Intersection Points P and Q:**
The line \( AL \) can be represented parametrically as:
\[ \vec{R} = \vec{A} + t(\vec{L} - \vec{A}) \]
The line \( BD \) can be represented as:
\[ \vec{R} = \vec{B} + s(\vec{D} - \vec{B}) \]
Setting these equal to find \( P \):
\[ \vec{A} + t\left(\vec{A} + \vec{a} + \frac{1}{3}\vec{b} - \vec{A}\right) = \vec{A} + \vec{a} + s\left(\vec{A} + \vec{b} - (\vec{A} + \vec{a})\right) \]

Similarly, for point \( Q \) on line \( AM \):
\[ \vec{R} = \vec{A} + u(\vec{M} - \vec{A}) \]

5. **Calculate the Ratios:**
From the calculations above, we find that \( P \) divides \( DB \) in the ratio \( 3:1 \) and \( Q \) divides \( DB \) in the ratio \( 1:3 \).

6. **Find the Lengths:**
The length \( PQ \) can be expressed in terms of \( DB \):
\[ PQ = \frac{1}{2} DB \quad \text{(since \( PQ \) is half of \( DB \))} \]

7. **Final Ratio:**
Therefore, the ratio \( PQ : DB \) is:
\[ PQ : DB = \frac{1}{2} : 1 = 1 : 2 \]

### Conclusion:
The final answer is:
\[ PQ : DB = 1 : 2 \]