A circle equation in center-radius form tells you where the center is and gives the radius squared. To test a point, square its horizontal and vertical changes from the center, then add. Watch the signs inside the parentheses: copying them as the center’s coordinates can make an otherwise promising point miss the circle.
Hints
- Hint 1
In center-radius form, the center’s coordinates make both parentheses zero. What value of makes , and what value of makes ? Check the signs before looking at the points.
- Hint 2
A point is on the circle when its squared horizontal and vertical changes add to the number on the right. Square and , then compare their sum with .
Step-by-step
Measure from the center
Step 1Read the circle’s center
No Desmos needed. The circle is already in center-radius form, and the -- right-triangle relationship gives the distance directly. In center-radius form, , the center is . Here , so the center is , not . Measure each point’s changes from that center, not from the origin.
- Step 2
Find changes that reach the circle
The Pythagorean theorem says the squares of a point’s horizontal and vertical changes add to its distance from the center squared. The choices contain and . Square and add:
That matches the circle’s squared radius. Either number can be the horizontal change, as long as the point’s coordinates put it the right distance from the center.
- Step 3
Match the changes to a point
Start at and go right and up. That gives . Check it in the given equation:
The and terms cancel, so this point lies on the circle regardless of their values. Choice C.