Question

If p and q are nonzero integers, which pair of points must lie in the same quadrant?
A. (p, q) and (q, p)
B. (p, q) and (2p, 2q)
C. (p, q) and (–p, –q)
D. (p, q) and (p – 2, q – 2)

Answer 1

To determine which pair of points must lie in the same quadrant, let's analyze each option one by one.

**A. (p, q) and (q, p)**
- Both points will lie in the same quadrant if both \( p \) and \( q \) are either both positive or both negative. However, if \( p \) is positive and \( q \) is negative (or vice versa), the points will be in different quadrants.
- So this option is not guaranteed to lie in the same quadrant.

**B. (p, q) and (2p, 2q)**
- The point (2p, 2q) is obtained by multiplying both coordinates of (p, q) by 2. Thus, if \( p \) and \( q \) are both positive or both negative, (2p, 2q) will also be in the same quadrant as (p, q).
- So this pair will always lie in the same quadrant.

**C. (p, q) and (–p, –q)**
- The point (–p, –q) is in the opposite quadrant of (p, q). For positive \( p \) and \( q \), (p, q) is in the first quadrant and (–p, –q) is in the third quadrant. If \( p \) and \( q \) are negative, (p, q) is in the third quadrant and (–p, –q) is in the first quadrant.
- Thus, this option cannot be guaranteed to lie in the same quadrant.

**D. (p, q) and (p − 2, q − 2)**
- Subtracting 2 from both coordinates changes the positioning of the point but does not guarantee that it remains in the same quadrant. For example, if \( p \) and \( q \) are both positive but less than or equal to 2, the resulting point could lie in a different quadrant.
- Therefore, this option is not guaranteed to lie in the same quadrant.

Among the options analyzed, the only pair that must lie in the same quadrant is:

**B. (p, q) and (2p, 2q)**