Consider the following statements in respect of the points (p,p-3),(q+3,q) and (6,3) :
1. The points lie on a straight line.
2. The points always lie in the first quadrant only for any value of p and q.
Which of the above statements is/are correct?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Answer

To determine the correctness of the given statements regarding the points \( (p, p-3) \), \( (q+3, q) \), and \( (6, 3) \), we will analyze each statement step by step.

### Step 1: Check if the points lie on a straight line

1. **Identify the points:**
- Point A: \( (p, p-3) \)
- Point B: \( (q+3, q) \)
- Point C: \( (6, 3) \)

2. **Calculate the slopes between the points:**
- **Slope of AB**:
\[ \text{slope}_{AB} = \frac{(q - (p - 3))}{((q + 3) - p)} = \frac{q - p + 3}{q + 3 - p} \]

- **Slope of BC**:
\[ \text{slope}_{BC} = \frac{(3 - q)}{(6 - (q + 3))} = \frac{3 - q}{3 - q} = 1 \]

- **Slope of AC**:
\[ \text{slope}_{AC} = \frac{(3 - (p - 3))}{(6 - p)} = \frac{3 - p + 3}{6 - p} = \frac{6 - p}{6 - p} = 1 \]

3. **Compare the slopes:**
- For the points to be collinear, the slopes must be equal. We find that:
\[ \text{slope}_{AB} = \text{slope}_{BC} = \text{slope}_{AC} = 1 \]
- Since all slopes are equal, the points lie on a straight line.

### Conclusion for Statement 1:
- **Statement 1 is correct.**

### Step 2: Check if the points always lie in the first quadrant

1. **Analyze the coordinates of each point:**
- For point A \( (p, p-3) \):
- The x-coordinate \( p \) must be positive.
- The y-coordinate \( p-3 \) must also be positive, which implies \( p > 3 \).
- For point B \( (q+3, q) \):
- The x-coordinate \( q+3 \) is always positive for any \( q \).
- The y-coordinate \( q \) must be positive, which implies \( q > 0 \).
- For point C \( (6, 3) \):
- Both coordinates are positive, so this point lies in the first quadrant.

2. **Determine the conditions for points A and B:**
- Point A lies in the first quadrant if \( p > 3 \).
- Point B lies in the first quadrant if \( q > 0 \).

3. **Conclusion about the conditions:**
- If \( p \) is less than or equal to 3, point A will not lie in the first quadrant.
- Therefore, the points do not always lie in the first quadrant for any value of \( p \) and \( q \).

### Conclusion for Statement 2:
- **Statement 2 is incorrect.**

### Final Answer:
- **Only Statement 1 is correct.**

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