Two points on a circle must be the same distance from its center. When they have the same -coordinate, their perpendicular bisector, the line through their midpoint at a right angle, is vertical. Graph it with the given center line in Desmos to find the center, then find the squared radius. The tempting trap is treating the two points as diameter endpoints when the question never says they are.
Hints
- Hint 1
Both points have the same height. For the center to be equally far from them, its -coordinate must be halfway between theirs. Their midpoint gives you one coordinate of the center, not necessarily the whole center.
- Hint 2
The center must also lie on the given line. The intersection, where that line meets the vertical line through the points’ midpoint, gives the center’s location.
- Hint 3
A radius runs from the center to a point on the circle. Square the horizontal and vertical changes to either given point and add them. The equation needs this squared radius, not the radius itself.
Step-by-step
Find the center, then the squared radius
Step 1Find the center’s horizontal coordinate
A circle’s center is equally far from and . Because the points have the same height, the center must sit halfway between their -coordinates. Find that coordinate:
The points aren’t stated to be endpoints of a diameter, so their midpoint fixes only the center’s horizontal location. The center lies somewhere on .
- Step 2
Locate the center on the given line
The center is on both and the given line . Type both equations into Desmos and click their intersection, the point on both lines. Desmos shows , so that is the center.
- Step 3
Find the squared radius
A radius reaches from the center to either given point. Using , square the horizontal and vertical changes and add them. Type into Desmos. It prints , so the squared radius is .
- Step 4
Write the circle’s equation
The standard form uses center and the squared radius. Put in and :
The minus signs place the center at , so this is the circle described. Choice D.