Circles

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Circle: Centre + Radius, Completed Square

(x-a)^2 + (y-b)^2 = r^2

\text{centre } (a,b), \quad \text{radius } r

Centre and radius read straight off the equation.

Reflex: Need exactly two ingredients — centre and radius — to write a circle's equation.

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6 worked examples

(x-a)^2 + (y-b)^2 = r^2

\text{centre } (a,b), \quad \text{radius } r

Centre and radius read straight off the equation.

Reflex: Need exactly two ingredients — centre and radius — to write a circle's equation.

\text{Centre} = \left(\frac{\text{coeff of } x}{-2}, \ \frac{\text{coeff of } y}{-2}\right)

r = \sqrt{a^2 + b^2 - c}

Centre hides in the x,y coefficients; radius needs its own formula.

Reflex: Divide by MINUS 2 — the sign flips back.

Sub the line into the circle, solve the quadratic. Two roots = secant, one = tangent, none = miss.

Reflex: Intersection points → simultaneous equations, every time.

The perpendicular bisector of any chord passes through the centre. Two points on a circle → their bisector is a line the centre must lie on.

Reflex: Draw two chords, bisect both — intersection IS the centre.

A tangent is perpendicular to the radius at the point of contact — so the normal at any point passes through the centre.

Reflex: Given a tangent, drop a perpendicular from the centre — its foot is the contact point, its length the radius.

d > r \Rightarrow \text{outside}, \quad d < r \Rightarrow \text{inside}, \quad d = r \Rightarrow \text{on circle}

Compare distance to centre with radius. Reflex: One distance computation decides it — skip the equation.