Rationale
y=2/3 x represents the equation of the line passing through points A and B.
The line through points A and B, which are endpoints of the diameter of a circle centered at (0, 0), has a slope of 2/3, giving the equation y=2/3 x. This slope indicates that for every 3 units moved horizontally, the line moves up 2 units vertically.
A) The equation y=-2/3 x implies a negative slope, which would indicate that the line descends as it moves from left to right. However, since points A and B are symmetric about the origin (0, 0) and AB forms a diameter, this line cannot represent the correct orientation of the diameter.
B) The equation y=2/3 x accurately reflects a positive slope, indicating the line rises as it moves from left to right. This slope corresponds to the relationship between the coordinates of points A and B, confirming that they indeed lie on this line as endpoints of the circle's diameter.
C) The equation y=3/2 x suggests a steeper slope than the actual line through A and B. This would imply that for every 2 units moved horizontally, the line rises 3 units, which does not represent the correct slope resulting from the coordinates of points A and B.
D) The equation y=4x indicates a very steep slope, suggesting a significant rise with minimal horizontal movement. This would not accurately depict the line through points A and B, as the slope of 4 is greater than what the coordinates of the diameter endpoints would allow.
Conclusion
The line passing through points A and B, the diameter of a circle centered at the origin, is represented by the equation y=2/3 x. This equation captures the correct slope based on the geometry of the circle and the location of the points, while the other choices reflect incorrect slopes that do not correspond to the line's actual orientation.
