9709 Pure 1 · 1.3 Coordinate Geometry · Sessions 1 and 2
Every point you drop is colored by a distance test. Green means it passes. Drop enough points and the green ones line up by themselves. Then the algebra shows why one test gives a straight line and the other gives a circle.
Tap or drag on the grid to paint points. Drag A or B to move them, or tab to a point and use the arrow keys.
Where do you think the green points will be?
Call any green point P(x, y). It is the same distance from A and B, so PA = PB. Square both sides to get rid of the square roots, then expand.
Tap or drag on the grid to paint points. Drag C to move it, or tab to it and use the arrow keys.
Where do you think the green points will be?
Call any green point P(x, y). Its distance from C is r. Write that distance with the distance formula, then square both sides.
One idea, two answers. Both tests start from the distance formula, and both times you square it.
With A and B there is a distance on each side of the equation. Each side has its own x² and y², so they cancel. Only x and y terms are left, and that is a straight line.
With C there is just one distance, so nothing cancels. The x² + y² stays in the equation, and that is why you get a circle: (x − a)² + (y − b)² = r².