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Learn Index · Geometry and Spatial Reasoning Grade 11–12

Equations of circles

(x − h)² + (y − k)² = r² is a circle with centre (h, k) and radius r.

Read

From distance to equation

A circle is all points (x, y) at distance r from a centre (h, k). Using the distance formula:

√[(x − h)² + (y − k)²] = r

Squaring gives the standard form:

(x − h)² + (y − k)² = r²

Circles centred at the origin

x² + y² = r²

Example: x² + y² = 25 → centre (0, 0), radius 5.

Reading standard form

Equation Centre Radius
(x − 3)² + (y − 2)² = 16 (3, 2) 4
(x + 1)² + (y − 5)² = 9 (−1, 5) 3
x² + (y + 4)² = 7 (0, −4) √7

Watch the signs: (x + 1) means h = −1.

Writing an equation

Example: Centre (2, −3), radius 6

(x − 2)² + (y + 3)² = 36

Example: Centre (1, 1), passing through (4, 5)

  • r² = (4 − 1)² + (5 − 1)² = 9 + 16 = 25
  • (x − 1)² + (y − 1)² = 25

Example: Diameter from (−2, 3) to (6, 9)

  • Centre (midpoint): (2, 6)
  • r = half of √(64 + 36) = 5
  • (x − 2)² + (y − 6)² = 25

General form

Expanding gives the general form:

x² + y² + Dx + Ey + F = 0

General form → standard form

Use completing the square.

Example: x² + y² − 6x + 4y − 12 = 0

  1. Group: (x² − 6x) + (y² + 4y) = 12
  2. Complete: (x² − 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4
  3. (x − 3)² + (y + 2)² = 25 → centre (3, −2), radius 5

Is a point on, inside or outside?

Substitute the point into the left side:

  • = r² → on the circle
  • < r² → inside

  • r² → outside

Example: Is (1, 2) inside x² + y² = 9? 1 + 4 = 5 < 9 → inside.

Applications

  • Radio and cell tower coverage zones
  • Earthquake epicentre location (intersecting circles)
  • Circular tracks, ripples and orbits

Worked examples

Example 1: Centre and radius of (x − 7)² + y² = 49? (7, 0), 7.

Example 2: Equation: centre (−4, 0), radius 2? (x + 4)² + y² = 4.

Example 3: Is (5, 5) on x² + y² = 50? 25 + 25 = 50 → yes.

Common mistakes

  • Using r instead of r² on the right side.
  • Wrong signs for h and k.
  • Forgetting to add to both sides when completing the square.

Try it yourself

  1. Centre and radius of (x + 2)² + (y − 6)² = 81?
  2. Equation with centre (3, −1) and radius 4.
  3. Convert x² + y² + 8x − 2y + 8 = 0 to standard form.
  4. Is (3, 4) inside, on or outside x² + y² = 20?

Answers: 1) (−2, 6), 9 2) (x − 3)² + (y + 1)² = 16 3) (x + 4)² + (y − 1)² = 9 4) outside (25 > 20)

Quick recap

  • Standard form: (x − h)² + (y − k)² = r², centre (h, k), radius r.
  • Expand to get general form; use completing the square to go back.
  • Substitute points to test on/inside/outside.