Question

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?

1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2

Answer 1

Correct Answer - Option 1 : 1 only

CONCEPT:

The three points A (x₁, y₁), B (x₂, y₂) and C (x₃, y₃) are said to be collinear if the area of Δ ABC is zero i.e \(\left| {\begin{array}{*{20}{c}} {{x_1}}&{{y_1}}&1\\ {{x_2}}&{{y_2}}&1\\ {{x_3}}&{{y_3}}&1 \end{array}} \right| = 0\)

CALCULATION:

Given: (p, p - 3), (q + 3, q) and (6, 3) are three points in a 2D plane.

Statement 1: The points lie on a straight line.

As we know that, if three points A (x1, y1), B (x2, y2) and C (x3, y3) are said to be collinear if the area of Δ ABC is zero i.e \(\left| {\begin{array}{*{20}{c}} {{x_1}}&{{y_1}}&1\\ {{x_2}}&{{y_2}}&1\\ {{x_3}}&{{y_3}}&1 \end{array}} \right| = 0\)

Let A = (p, p - 3), B = (q + 3, q) and C = (6, 3)

⇒ \(\left| {\begin{array}{*{20}{c}} {{p}}&{{p-3}}&1\\ {{q+3}}&{{q}}&1\\ {{6}}&{{3}}&1 \end{array}} \right| = p (q - 3) - (p-3)(q+3 - 6) + 1(3q+9-6q) \)

⇒ \(\left| {\begin{array}{*{20}{c}} {{p}}&{{p-3}}&1\\ {{q+3}}&{{q}}&1\\ {{6}}&{{3}}&1 \end{array}} \right| = 0\)

So, as we can see that area of triangle ABC = 0 for any value of p and q.

Hence, statement 1 is true.

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

As we can see that, for any value of p and q it is not necessary that the points lies in the first quadrant only.

Hence, statement 2 is false.

Hence, the correct option is 1.