Two points on a circle aren't automatically the ends of a diameter. First check whether both lie on the given line through the center. If they do, use Desmos to find the distance between them, then halve it for the radius. The tempting mistake is to use the full distance as the radius.
Hints
- Hint 1
A diameter goes through the center and reaches the circle on opposite sides. Check whether both given points satisfy the line's equation. If they do, what does the line tell you about the segment between them?
- Hint 2
The distance formula uses the horizontal and vertical changes between two points to find their distance. Here that distance runs across the whole circle. How much of it is one radius?
Step-by-step
Recognize the diameter, then halve its length
Step 1Check the center's line
Type , , and in Desmos. Both plotted points sit on the line: their coordinates give and . The problem also puts the center on that line, so the center and both circle points are in a straight line.
- Step 2
Identify what the two points span
A line through a circle's center reaches the circle on opposite sides. Since these are two points on that line and on the circle, they are the ends of a diameter, the segment across the circle through its center. Two circle points alone wouldn't guarantee a diameter; the given center line is what makes this work.
- Step 3
Find the full diameter
Add in Desmos. It shows about , the diameter's length. For its exact form, the horizontal and vertical changes are both ( and ). Use the distance formula to square those changes and add them:
Add the squares:
Simplify the square root:
- Step 4
Halve the diameter for the radius
Add in Desmos. It shows about because a radius is half a diameter. To match the exact form, halve the length you found:
Divide by :
The circle's radius is . Choice D.