Exams
Topic module

Circles, Tangency and Intersections

Read and form circle equations, use radius–tangent geometry and solve line–circle or circle–circle intersections.

Long-form learning
Concept to Risk to Memory to Check-up

How to study Higher Mathematics

Practise a skill without a calculator first, connect it to neighbouring techniques, then apply it in unfamiliar contexts with complete working and interpretation.

Core concepts

Concept 1

centre-radius form

Exam cue: Can the equation be completed to squares?

Concept 2

general circle form

Exam cue: Where does the radius meet the tangent?

Concept 3

radius

Exam cue: How many intersections should the algebra produce?

Concept 4

tangent property

Concept 5

simultaneous intersections

Risk pitfalls and guardrails

Reading the centre signs directly from (x − a)² + (y − b)²

Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.

Forgetting the tangent is perpendicular to the radius

Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.

Discarding an intersection root without checking

Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.

Memory anchors

Centre signs reverse

In (x − a)² + (y − b)² = r², the centre is (a, b).

General centre is (−g, −f)

Use the supplied x² + y² + 2gx + 2fy + c = 0 form.

Radius is positive

For the general form, r = √(g² + f² − c).

Tangent is radius-perpendicular

At the contact point, their gradients multiply to −1 when defined.

Intersections are simultaneous

Substitute or eliminate between the relevant equations.

Discriminant counts contact

A repeated solution can indicate tangency.

Checkpoint rule

Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.

Knowledge Check (after reading)

Short check-up to confirm understanding of this module.

Check-up Questions

1-2 question checkpoint

Find the centre and radius of (x−3)²+(y+2)²=25.

Find the equation of the circle with centre (−4,1) and radius 3.

Answer all questions to submit.

Next step personalized recommendations

Continue learning

Move forward only after this module is stable.

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