A circle through two points with its center restricted to an axis calls for squared distance: both points must be equally far from the center. Write the center with one unknown coordinate, then use a Desmos regression to make the two squared distances equal. Find the squared radius from either point and write the circle in standard form. Don’t use the points’ midpoint unless they’re identified as diameter endpoints.
Hints
- Hint 1
The -axis is the line . If you call the center’s unknown -coordinate , its coordinates are . What must be true of the distances from this center to the two given points?
- Hint 2
A point’s squared distance from is its horizontal change squared plus its vertical change squared. Set that expression equal for the two points. Don’t use their midpoint unless you know they’re diameter endpoints.
- Hint 3
Once you know the center, a radius reaches from it to either given point. Square the horizontal and vertical changes and add them. That gives , which belongs on the right side of the circle equation.
Step-by-step
Use equal distances from the center
Step 1Name the unknown center
The -axis is the line , so the center’s -coordinate is . Call its unknown -coordinate ; the center is .
- Step 2
Set the two squared distances equal
A radius runs from the center to any point on the circle. Both given points are equally far from , so square and add their horizontal and vertical changes: . The midpoint would be the center only if these points were ends of a diameter.
- Step 3
Find the center’s height
Type the distance equation with instead of so Desmos fits the unknown : . Under PARAMETERS, Desmos shows , so the center is .
- Step 4
Find the squared radius
Use the center and to find the squared radius. Type beneath the regression. Desmos prints , so ; you don’t need to take a square root.
- Step 5
Write the circle equation
In standard form, . Insert center and : . Simplify : . This equation describes the circle in the question. Choice A.