A centre is equidistant from every point on its circle. A diameter has its midpoint at the centre and is twice the radius. Method: use midpoint for a diameter, or equate squared distances when the centre is unknown. Combine this with any given line containing the centre. Then substitute the centre and one point into the distance formula. Example: a centre (h,k) equidistant from (0,0) and (4,0) satisfies h2+k2=(h-4)2+k2, so h=2. If k=h+1, then C=(2,3) and r2=13. Exam check: two points alone do not usually determine a unique circle; a diameter condition or another independent constraint is needed. Use half the diameter, not the whole diameter, for the radius.

A centre is equidistant from every point on its circle. A diameter has its midpoint at the centre and is twice the radius.

Method: use midpoint for a diameter, or equate squared distances when the centre is unknown. Combine this with any given line containing the centre. Then substitute the centre and one point into the distance formula.

Example: a centre \((h,k)\) equidistant from \((0,0)\) and \((4,0)\) satisfies \(h^2+k^2=(h-4)^2+k^2\), so \(h=2\). If \(k=h+1\), then \(C=(2,3)\) and \(r^2=13\).

Exam check: two points alone do not usually determine a unique circle; a diameter condition or another independent constraint is needed. Use half the diameter, not the whole diameter, for the radius.

Constructing circles from geometric data

Constructing circles from geometric data