The Circle Topic Review summary

Key learning points

• $(x-h)^2+(y-k)^2=r^2$ has centre $(h,k)$ and radius $r$.
• General form $x^2+y^2+2gx+2fy+c=0$ → centre $(-g,-f)$, $r=\sqrt{g^2+f^2-c}$.
• Line meets circle: substitute; discriminant for 0/1/2 intersections.
• Tangent ⊥ radius at the point of contact.
• Angle in a semicircle / chord properties can mix with coordinates.

Worked example 1 — Centre–radius

$x^2+y^2-4x+6y-3=0$.

Working

Complete the square.

Worked example 2 — Tangent

$y=2x+c$ tangent to $x^2+y^2=5$.

Working

Discriminant zero.

Worked example 3 — (x-h)^2+(y-k)^2=r^2 has centre (h,k) and r

$(x-h)^2+(y-k)^2=r^2$ has centre $(h,k)$ and radius $r$.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — General form x^2+y^2+2gx+2fy+c=0 → centre

General form $x^2+y^2+2gx+2fy+c=0$ → centre $(-g,-f)$, $r=\\sqrt{g^2+f^2-c}$.

Working

Attempt without notes, then check the answers PDF.

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.