The Circle summary

Key learning points

• $(x-h)^2+(y-k)^2=r^2$ has centre $(h,k)$ and radius $r$.
• Expanded form: complete the square for $x$ and $y$.
• Line–circle intersections: substitute and solve a quadratic.
• Geometric facts (tangent ⊥ radius) often sit alongside coordinates.

Worked example 1 — Centre and radius

Find centre and radius of $x^2+y^2-6x+4y-12=0$.

Working

Centre $(3,-2)$, radius 5.

Worked example 2 — has centre and radius

Use this method from the The Circle sheet: $(x-h)^2+(y-k)^2=r^2$ has centre $(h,k)$ and radius $r$.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 3 — Expanded form: complete the square for and

Use this method from the The Circle sheet: Expanded form: complete the square for $x$ and $y$.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 4 — Line–circle intersections: substitute and solve a quadr

Use this method from the The Circle sheet: Line–circle intersections: substitute and solve a quadratic.

Working

Each key learning point matches a question type on the printable worksheet.

Worked example 5 — Geometric facts (tangent ⊥ radius) often sit alongside

Use this method from the The Circle sheet: Geometric facts (tangent ⊥ radius) often sit alongside coordinates.

Working

Each key learning point matches a question type on the printable worksheet.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.