Points A and B are both in the line segment PQ and on the same side of its midpoint. A divides PQ in the ratio `2 : 3`, and B divides PQ in the ratio `3 : 4`. If AB=2, then the length of PQ is :

Options

A. 70
B. 75
C. 80
D. 85

Answer

To solve the problem, we need to find the length of the line segment PQ given the ratios in which points A and B divide it, and the distance between points A and B.

### Step-by-Step Solution:

1. **Understanding the Ratios**:
- Point A divides PQ in the ratio 2:3. This means if we let the total length of PQ be divided into 5 equal parts (2 + 3), then:
- Length from P to A = \( \frac{2}{5} \) of PQ
- Length from A to Q = \( \frac{3}{5} \) of PQ
- Point B divides PQ in the ratio 3:4. This means if we let the total length of PQ be divided into 7 equal parts (3 + 4), then:
- Length from P to B = \( \frac{3}{7} \) of PQ
- Length from B to Q = \( \frac{4}{7} \) of PQ

2. **Expressing Lengths in Terms of PQ**:
- Let the length of PQ be \( x \).
- Therefore:
- Length from P to A = \( \frac{2}{5}x \)
- Length from A to Q = \( \frac{3}{5}x \)
- Length from P to B = \( \frac{3}{7}x \)
- Length from B to Q = \( \frac{4}{7}x \)

3. **Finding the Distance AB**:
- The distance AB can be expressed as:
\[ AB = |PB - PA| = \left| \frac{3}{7}x - \frac{2}{5}x \right| \]
- To calculate this, we need a common denominator, which is 35:
\[ PB = \frac{3}{7}x = \frac{15}{35}x \]
\[ PA = \frac{2}{5}x = \frac{14}{35}x \]
- Thus,
\[ AB = \left| \frac{15}{35}x - \frac{14}{35}x \right| = \left| \frac{1}{35}x \right| = \frac{1}{35}x \]

4. **Setting Up the Equation**:
- We know from the problem statement that AB = 2:
\[ \frac{1}{35}x = 2 \]

5. **Solving for x**:
- Multiply both sides by 35:
\[ x = 2 \times 35 = 70 \]

6. **Conclusion**:
- The length of PQ is \( 70 \) units.

### Final Answer:
The length of PQ is **70**.