Labs / Mathematics
Distance & Locus
A circle is not just a round shape — it is every point that sits the same distance from a center. Drag the center, set a radius, and move the test point P around to watch its distance change and see exactly when it lands inside, on, or outside the circle.
eq x² + y² = 9|CP| 4.47 uP is OUTSIDE
What to try
- Where does P have to be for its distance to exactly equal the radius — and how many such points are there?
- Slide the center off the origin. How do the numbers inside the equation (x − h)² + (y − k)² = r² change?
- Turn on the distance formula. Can you see how Δx and Δy build a right triangle whose hypotenuse is |CP|?
- Keep P fixed and grow the radius. At what value does P flip from outside to inside?
- Can you place C and P so that |CP| is a whole number even though neither Δx nor Δy is zero?