21. In the xy-plane above, the circle has center (0, 0) and AB is a diameter of the circle. What is the equation of the line passing through points A and B?
Answer: B
The equation of the line passing through points A and B is y=2/3 x.
The line that passes through points A and B, which are endpoints of a diameter of a circle centered at (0, 0), has a slope of 2/3, leading to the equation y=2/3 x.
A) y=-2/3 x
This option represents a line with a negative slope of -2/3. Since points A and B are on the same diameter and the line should pass through the center (0, 0), a negative slope would not be consistent with the expected position of the points in this context.
B) y=2/3 x
This option correctly represents a line with a positive slope of 2/3, which is consistent with the geometry of a diameter in the circle centered at (0, 0). If points A and B are positioned in the first and third quadrants, this equation accurately reflects the direction of the line.
C) y=3/2 x
This option indicates a line with a steeper positive slope of 3/2. While it is a valid linear equation, it does not correspond to the angle or position of a line passing through a diameter of a circle centered at the origin, thus making it incorrect.
D) y=4x
This option presents a line with a very steep slope of 4. Similar to option C, this slope does not align with the expected slope of the diameter line through the circle and is therefore not applicable in this context.
Conclusion
The equation y=2/3 x accurately describes the line through points A and B, as it reflects the correct positive slope consistent with the circle's diameter. All other options fail to meet the geometric requirements of the problem, whether by having incorrect slopes or orientation. Thus, option B is definitively the correct choice.