Coordinate Systems Topic Review summary

Key learning points

• Standard parabola $y^2=4ax$ with focus $(a,0)$ and directrix $x=-a$.
• Parametric points $(at^2,2at)$ on $y^2=4ax$.
• Ellipse/hyperbola standard forms and key intercepts when on the syllabus.
• Tangents and normals via calculus or standard formulae.
• Convert between Cartesian and parametric representations.

Worked example 1 — Focus

Focus of $y^2=8x$.

Working

$4a=8\Rightarrow a=2$.

Worked example 2 — Parametric

Point with parameter $t=3$ on $y^2=4ax$.

Working

$(at^2,2at)$.

Worked example 3 — Practice point 1

Standard parabola $y^2=4ax$ with focus $(a,0)$ and directrix $x=-a$.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — Practice point 2

Parametric points $(at^2,2at)$ on $y^2=4ax$.

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.