Question
If j and k are nonzero integers, which pair of points must lie in the same quadrant?
(j, j) and (k, k)
(j, k) and (jk, jk)
(j + k, 3) and (3, j + k)
(3j, 3k) and (StartFraction 3 Over j EndFraction, StartFraction 3 Over k EndFraction)
(j, j) and (k, k)
(j, k) and (jk, jk)
(j + k, 3) and (3, j + k)
(3j, 3k) and (StartFraction 3 Over j EndFraction, StartFraction 3 Over k EndFraction)
Answers
GPT-4o mini
To determine which pair of points must lie in the same quadrant, we need to analyze each option based on the condition that \( j \) and \( k \) are nonzero integers.
1. **Points (j, j) and (k, k)**:
- Both points (j, j) and (k, k) will lie in the same quadrant if both \( j \) and \( k \) are either positive or both are negative.
- Thus, it is possible for them to be in different quadrants if one is positive and the other is negative.
2. **Points (j, k) and (jk, jk)**:
- The point (j, k) will be in the same quadrant if both \( j \) and \( k \) are positive or both are negative.
- The point (jk, jk) will always be positive if \( j \) and \( k \) are both positive or both are negative because the product of two negatives is positive.
- Therefore, if \( j \) and \( k \) have different signs, this will produce points that are definitely in different quadrants.
3. **Points (j + k, 3) and (3, j + k)**:
- The point (j + k, 3) can be in any quadrant depending on the signs of \( j \) and \( k \).
- Although the point (3, j + k) can also vary as it will depend on the sign of \( j + k \).
- Thus, these points can also lie in different quadrants.
4. **Points (3j, 3k) and ( \( \frac{3}{j} \), \( \frac{3}{k} \) )**:
- The point (3j, 3k) will be in the same quadrant as both \( j \) and \( k \) will determine the quadrant based on their signs being positive or negative. Since they are multiplied by 3, they will always retain the sign of \( j \) and \( k \).
- The point \( ( \frac{3}{j}, \frac{3}{k} ) \) will also be in the same quadrant as the signs of \( j \) and \( k \) will determine the signs of the fractions (3/j and 3/k will be positive if \( j \) and \( k \) are positive or both negative).
- Therefore, both points will always lie in the same quadrant.
Based on this analysis, the pair of points that must lie in the same quadrant is:
**(3j, 3k) and ( \( \frac{3}{j} \), \( \frac{3}{k} \) )**.
1. **Points (j, j) and (k, k)**:
- Both points (j, j) and (k, k) will lie in the same quadrant if both \( j \) and \( k \) are either positive or both are negative.
- Thus, it is possible for them to be in different quadrants if one is positive and the other is negative.
2. **Points (j, k) and (jk, jk)**:
- The point (j, k) will be in the same quadrant if both \( j \) and \( k \) are positive or both are negative.
- The point (jk, jk) will always be positive if \( j \) and \( k \) are both positive or both are negative because the product of two negatives is positive.
- Therefore, if \( j \) and \( k \) have different signs, this will produce points that are definitely in different quadrants.
3. **Points (j + k, 3) and (3, j + k)**:
- The point (j + k, 3) can be in any quadrant depending on the signs of \( j \) and \( k \).
- Although the point (3, j + k) can also vary as it will depend on the sign of \( j + k \).
- Thus, these points can also lie in different quadrants.
4. **Points (3j, 3k) and ( \( \frac{3}{j} \), \( \frac{3}{k} \) )**:
- The point (3j, 3k) will be in the same quadrant as both \( j \) and \( k \) will determine the quadrant based on their signs being positive or negative. Since they are multiplied by 3, they will always retain the sign of \( j \) and \( k \).
- The point \( ( \frac{3}{j}, \frac{3}{k} ) \) will also be in the same quadrant as the signs of \( j \) and \( k \) will determine the signs of the fractions (3/j and 3/k will be positive if \( j \) and \( k \) are positive or both negative).
- Therefore, both points will always lie in the same quadrant.
Based on this analysis, the pair of points that must lie in the same quadrant is:
**(3j, 3k) and ( \( \frac{3}{j} \), \( \frac{3}{k} \) )**.