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Circle equations and tangents

Use the equation of a circle centred at the origin and find the equation of a tangent at a point on the circle.

Geometry and MeasuresHigherGrades 7 to 9Skill

Curriculum path: GCSE Maths > Edexcel > Geometry and Measures > Circle equations and tangents

Pearson Edexcel GCSE Maths Higher algebra/geometry: recognise and use the equation of a circle centred at the origin and find tangent equations.

Practise this topicUpload your answerAsk AI to explain thisTry exam-style questions
Revision notes

Theory, examples, and quick checks.

Keep the method short, then practise straight away. This note is written for GCSE Maths Edexcel students who need clear working and reliable method marks.

Theory

A circle centred at the origin with radius r has equation x² + y² = r².

The equation comes from Pythagoras. Any point (x, y) on the circle makes a right triangle with horizontal length x, vertical length y and hypotenuse r.

If the equation is x² + y² = 25, then r² = 25 and the radius is 5.

To check whether a point lies on the circle, substitute its x and y coordinates into x² + y².

A tangent to a circle is perpendicular to the radius at the point of contact.

To find a tangent equation, find the gradient of the radius from the origin to the point, then use the negative reciprocal for the tangent gradient.

Higher note: this topic often appears in graph/algebra questions, but the key geometry fact is radius perpendicular to tangent.

Key ruleCircle centre origin: x² + y² = r². Tangent gradient is the negative reciprocal of the radius gradient.

Diagram guide

O(x, y)rcircle centre origin: x² + y² = r²
Circle equationEvery point on the circle satisfies x² + y² = r².
OPradiustangentradius and tangent are perpendicularmultiply perpendicular gradients to get -1
Tangent gradientThe radius and tangent meet at 90 degrees, so their gradients multiply to -1.

Worked examples

Find the radius

A circle has equation x² + y² = 49. Find its radius.

O(x, y)rcircle centre origin: x² + y² = r²
Example: radius from equationThe number on the right is r².
  1. Compare x² + y² = 49 with x² + y² = r².
  2. r² = 49.
  3. r = 7.

Answer: 7

Check a point

Does the point (3, 4) lie on the circle x² + y² = 25?

O(x, y)rcircle centre origin: x² + y² = r²
Example: point on circleSubstitute x = 3 and y = 4.
  1. x² + y² = 3² + 4².
  2. 3² + 4² = 9 + 16 = 25.
  3. The point lies on the circle.

Answer: Yes

Find a tangent equation

Find the equation of the tangent to x² + y² = 25 at the point (3, 4).

OPradiustangentradius and tangent are perpendicularmultiply perpendicular gradients to get -1
Example: tangent at a pointThe radius from the origin to (3, 4) is perpendicular to the tangent.
  1. Gradient of radius from (0, 0) to (3, 4) is 4/3.
  2. Tangent gradient is the negative reciprocal: -3/4.
  3. Use y - 4 = -3/4(x - 3).
  4. Simplify: y = -3/4x + 25/4.

Answer: y = -3/4x + 25/4

Common mistakes

  • Thinking x² + y² = 25 has radius 25 instead of 5.
  • Forgetting to square negative coordinates when checking a point.
  • Using the same gradient for the tangent as the radius.
  • Finding the negative reciprocal the wrong way round.
  • Writing a tangent equation that does not pass through the point of contact.
  • Mixing this with circles not centred at the origin.

Quick exercise

Try these before moving to the exam-style questions.

  1. Find the radius of x² + y² = 36.
    O(x, y)rcircle centre origin: x² + y² = r²
    Quick check: radiusThe right side is r².
  2. Does (5, 12) lie on x² + y² = 169?
    O(x, y)rcircle centre origin: x² + y² = r²
    Quick check: substituteCheck 5² + 12².
  3. The radius gradient is 2/5. What is the tangent gradient?
    OPradiustangentradius and tangent are perpendicularmultiply perpendicular gradients to get -1
    Quick check: perpendicular gradientUse the negative reciprocal.
  4. The radius gradient is -3. What is the tangent gradient?
    OPradiustangentradius and tangent are perpendicularmultiply perpendicular gradients to get -1
    Quick check: whole-number gradientWrite -3 as -3/1 first.
  5. What geometry fact connects the radius and tangent?
    OPradiustangentradius and tangent are perpendicularmultiply perpendicular gradients to get -1
    Quick check: theorem linkThe tangent touches at one point.
Exam-style questions

Practise the same skill at three levels.

These are original GCSE-style questions with mark schemes, common wrong answers, and AI marking guidance so feedback stays close to exam expectations.

Basic GCSE styleHigherNon-calculator2 marks

Write down the radius of the circle x² + y² = 64.

O(x, y)rcircle centre origin: x² + y² = r²
Question diagram: radiusThe equation is x² + y² = r².
circle equationradiushigher graphs
Standard exam styleHigherNon-calculator3 marks

Show that the point (-5, 12) lies on the circle x² + y² = 169.

O(x, y)rcircle centre origin: x² + y² = r²
Question diagram: point on circleSubstitute both coordinates carefully.
circle equationsubstitutionshow that
ChallengeHigherNon-calculator5 marks

Find the equation of the tangent to the circle x² + y² = 25 at the point (4, 3).

OPradiustangentradius and tangent are perpendicularmultiply perpendicular gradients to get -1
Question diagram: tangent equationFind the radius gradient, then the perpendicular tangent gradient.
tangent equationperpendicular gradienthigher algebra geometry