Analyzing Points on a Straight Line and in Quadrants

The question asks us to evaluate two statements regarding three given points: (p, p - 3), (q + 3, q), and (6, 3). Let's analyze each statement carefully using principles of coordinate geometry.

Statement 1: The points lie on a straight line.

To determine if three points are collinear (lie on the same straight line), we can use several methods. One common method is to check if the area of the triangle formed by these three points is zero. The area of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is given by the formula:

\( \text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| \)

For the given points (p, p - 3), (q + 3, q), and (6, 3), let's assign:

• (x₁, y₁) = (p, p - 3)
• (x₂, y₂) = (q + 3, q)
• (x₃, y₃) = (6, 3)

Now, substitute these values into the area formula:

\( \text{Area} = \frac{1}{2} |p(q - 3) + (q + 3)(3 - (p - 3)) + 6((p - 3) - q)| \)

Simplify the expression inside the absolute value:

\( p(q - 3) = pq - 3p \)

\( (q + 3)(3 - p + 3) = (q + 3)(6 - p) = q(6 - p) + 3(6 - p) = 6q - pq + 18 - 3p \)

\( 6(p - 3 - q) = 6p - 18 - 6q \)

Now, sum these simplified terms:

\( (pq - 3p) + (6q - pq + 18 - 3p) + (6p - 18 - 6q) \)

Group like terms:

\( (pq - pq) + (-3p - 3p + 6p) + (6q - 6q) + (18 - 18) \)

\( 0 + 0 + 0 + 0 = 0 \)

The expression inside the absolute value simplifies to 0. Therefore, the area of the triangle formed by the three points is:

\( \text{Area} = \frac{1}{2} |0| = 0 \)

Since the area of the triangle is always 0, regardless of the values of p and q, the three points are always collinear. Thus, Statement 1 is correct.

Statement 2: The points always lie in the first quadrant only for any value of p and q.

The first quadrant is defined by the region where the x-coordinate is strictly positive (x > 0) and the y-coordinate is strictly positive (y > 0). For all three points to *always* lie in the first quadrant for *any* value of p and q, the x and y coordinates of each point must satisfy the conditions x > 0 and y > 0 universally for all possible real values of p and q.

Let's examine each point:

• Point 1: (p, p - 3)

  For this point to be in the first quadrant, we need p > 0 AND p - 3 > 0.

  p - 3 > 0 implies p > 3.

  So, for this point to be in the first quadrant, we need p > 3. This condition is not true for *any* value of p. For example, if p = 1, the point is (1, 1 - 3) = (1, -2), which is in the fourth quadrant.

• Point 2: (q + 3, q)

  For this point to be in the first quadrant, we need q + 3 > 0 AND q > 0.

  q + 3 > 0 implies q > -3.

  So, for this point to be in the first quadrant, we need q > -3 AND q > 0. The combined condition is q > 0. This condition is not true for *any* value of q. For example, if q = -1, the point is (-1 + 3, -1) = (2, -1), which is in the fourth quadrant.

• Point 3: (6, 3)

  The coordinates are 6 and 3. Since 6 > 0 and 3 > 0, this point is always in the first quadrant.

Since Points 1 and 2 do not *always* lie in the first quadrant for *any* value of p and q, Statement 2 is incorrect.

Conclusion

Based on our analysis, Statement 1 is correct, and Statement 2 is incorrect. Therefore, only Statement 1 is correct.

Revision Table: Coordinate Geometry Concepts

Additional Information: Collinearity and Quadrants

Understanding collinearity is fundamental in coordinate geometry. Besides the area method used in the solution, you can also check for collinearity by comparing slopes. The slope between the first two points should be equal to the slope between the second and third points. If all slopes are equal (and defined), the points are collinear. If any pair of points is the same, they are trivially collinear with any third point.

The concept of quadrants helps locate points in the Cartesian plane. Each quadrant is defined by the signs of the x and y coordinates. It's important to remember the strict inequalities (> 0 or < 0) for points lying *within* a quadrant. Points lying on the axes do not belong to any quadrant.

In this problem, the algebraic structure of the points (p, p-3) and (q+3, q) means their position relative to the first quadrant boundaries (x=0 and y=0) depends directly on the specific values of p and q. Point (6, 3) is a fixed point, always in the first quadrant, but the other points are variable based on p and q.