Practice Maths

L06 — Circles

Key Terms

  • Circle — the set of all points in the plane equidistant from a fixed centre point
  • Centre (h, k) — the fixed point from which all points on the circle are equal distance
  • Radius r — the constant distance from the centre to any point on the circle
  • Standard form — x² + y² = r² (centre at the origin)
  • Translated form — (x − h)² + (y − k)² = r² (centre at any point)
  • Completing the square — algebraic technique to convert the expanded form of a circle equation to standard form

Circle Equations

FormEquationCentreRadius
Standard (at origin)x² + y² = r²(0, 0)r
General (translated)(x − h)² + (y − k)² = r²(h, k)r
x y -4 -3 -2 -1 1 2 3 4 4 3 2 1 -1 -2 -3 -4 r = 3 (0,0) (3, 0) (-3, 0) (0, 3) (0, -3) x² + y² = 9
Circle x² + y² = 9 — centre (0, 0), radius 3

Key Features

Featurex² + y² = r²(x−h)² + (y−k)² = r²
Centre(0, 0)(h, k)
Radiusrr
Domain[−r, r][h−r, h+r]
Range[−r, r][k−r, k+r]
Is it a function?No — fails the vertical line test (each x near the centre gives two y-values)
Hot Tip: A circle is not a function — it fails the vertical line test. For any x strictly between the left and right edges of the circle, there are two y-values (one above the centre, one below). You can never write y = f(x) for a complete circle.

Completing the Square

To convert x² + y² + Ax + By + C = 0 to standard form:

  1. Group x-terms and y-terms: (x² + Ax) + (y² + By) = −C
  2. Add (A/2)² and (B/2)² to both sides.
  3. Factorise each group as a perfect square.
  4. Read off centre (h, k) and radius r = √(r²).
Example: x² + y² − 6x + 2y − 6 = 0
(x² − 6x + 9) + (y² + 2y + 1) = 6 + 9 + 1
(x − 3)² + (y + 1)² = 16
⇒ Centre (3, −1), radius 4.

What is a Circle?

A circle is the set of all points in the plane that are a fixed distance (the radius) from a fixed point (the centre). If the centre is at the origin and the radius is r, then any point (x, y) on the circle satisfies:

x² + y² = r²

This comes directly from the distance formula: the distance from (x, y) to (0, 0) is √(x² + y²), and we set this equal to r.

Worked Example 1: Write the equation and state key features

A circle has centre (0, 0) and radius 6. Write its equation and state the domain and range.

Solution:
Equation: x² + y² = 36
Domain: [−6, 6]    Range: [−6, 6]
The circle is not a function — for example, when x = 0, y = ±6 (two values).

Translated Circles

When the centre is moved to (h, k), the distance formula gives:

(x − h)² + (y − k)² = r²

Note the sign convention carefully: if the centre is at (−2, 3), the equation is (x + 2)² + (y − 3)² = r². The signs inside the brackets are opposite to the centre coordinates.

Worked Example 2: Identify centre and radius

State the centre and radius of each circle.
(a) (x − 4)² + (y + 3)² = 49     (b) x² + (y − 2)² = 5

Solution:
(a) Rewrite as (x − 4)² + (y − (−3))² = 49
    Centre: (4, −3),   Radius: r = √49 = 7

(b) Rewrite as (x − 0)² + (y − 2)² = 5
    Centre: (0, 2),   Radius: r = √5

Completing the Square

A circle equation is sometimes given in expanded form such as x² + y² − 4x + 6y + 4 = 0. To find the centre and radius, complete the square on both the x-terms and y-terms separately.

Method:

  1. Move the constant to the right: x² − 4x + y² + 6y = −4
  2. Complete the square on x: take half the x-coefficient (−2), square it (+4), add to both sides.
  3. Complete the square on y: take half the y-coefficient (3), square it (+9), add to both sides.
  4. Factorise and read off centre and radius.
Worked Example 3: Complete the square

Rewrite x² + y² − 4x + 6y + 4 = 0 in standard form. State the centre and radius.

Solution:
x² − 4x + y² + 6y = −4
(x² − 4x + 4) + (y² + 6y + 9) = −4 + 4 + 9
(x − 2)² + (y + 3)² = 9
⇒ Centre (2, −3), Radius 3

Is a Point Inside, On, or Outside the Circle?

Substitute the point (a, b) into the left-hand side of (x − h)² + (y − k)² = r² and compare with r²:

  • = r²: the point is on the circle
  • < r²: the point is inside the circle
  • > r²: the point is outside the circle
Worked Example 4: Point classification

For the circle (x − 1)² + (y − 2)² = 25, classify each point as inside, on, or outside.
(a) (4, 6)     (b) (1, 7)     (c) (5, 5)

Solution:
(a) (4−1)² + (6−2)² = 9 + 16 = 25 = r²  ⇒  On the circle
(b) (1−1)² + (7−2)² = 0 + 25 = 25 = r²  ⇒  On the circle
(c) (5−1)² + (5−2)² = 16 + 9 = 25 = r²  ⇒  On the circle

All three lie exactly on the circle with r = 5.

  1. Key features from equations. Fluency

    Complete the table.

    EquationCentreRadius
    x² + y² = 25
    (x − 3)² + (y − 2)² = 16
    (x + 1)² + (y − 4)² = 9
    x² + (y + 5)² = 1

  2. Write the equation. Fluency

    Write the equation of the circle with the given centre and radius.

    • (a) Centre (0, 0), radius 7
    • (b) Centre (2, −3), radius 5
    • (c) Centre (−4, 1), radius √3
    • (d) Centre (0, 6), radius 2

  3. Does the point lie on the circle? Fluency

    Determine whether each point lies on the circle x² + y² = 25. Substitute and check whether the equation holds.

    • (a) (3, 4)
    • (b) (0, 5)
    • (c) (1, 4)
    • (d) (−4, 3)

  4. Domain and range. Fluency

    State the domain and range for each circle. Write your answers in interval notation, e.g. [−5, 5].

    • (a) x² + y² = 36
    • (b) (x − 1)² + (y + 2)² = 25
    • (c) (x + 3)² + y² = 4
    • (d) x² + (y − 3)² = 9

  5. Completing the square. Understanding

    Rewrite each equation in standard form. State the centre and radius.

    • (a) x² + y² − 6x + 2y − 6 = 0
    • (b) x² + y² + 4x − 8y + 11 = 0
    • (c) x² + y² − 2x − 10y + 22 = 0
    • (d) x² + y² + 8x + 6y = 0

  6. Inside, on, or outside? Understanding

    The graph shows the circle (x − 2)² + (y − 3)² = 25 with four points labelled. For each point, first estimate from the graph whether it is inside, on, or outside the circle, then verify algebraically.

    x y −2 2 4 6 8 8 6 4 2 −2 (2, 3) A (2, 8) B (5, 3) C (7, 7) D (2, −2)
    • (a) Point A = (2, 8)
    • (b) Point B = (5, 3)
    • (c) Point C = (7, 7)
    • (d) Point D = (2, −2)

  7. Equation from centre and point. Understanding

    Find the equation of the circle with the given centre that passes through the given point. (First find r² using the distance formula, then write the equation.)

    • (a) Centre (0, 0), passes through (5, 12)
    • (b) Centre (3, −1), passes through (7, 2)
    • (c) Centre (−2, 4), passes through (1, 4)
    • (d) Centre (0, −3), passes through (4, 0)

  8. Mobile tower coverage. Understanding

    A mobile phone tower is at coordinates (3, 2) on a map (units in km). Its signal covers a circular area with radius 5 km.

    • (a) Write the equation of the boundary of the coverage area.
    • (b) Does the town at (6, 6) receive coverage?
    • (c) Does the house at (8, 3) receive coverage?
    • (d) A second tower at (−1, 2) has the same 5 km radius. Write its equation. Do the two coverage areas overlap? (Compare the distance between the two centres to the sum of the radii.)

  9. Endpoints of a diameter. Problem Solving

    Show that A(1, 7) and B(7, 1) are the endpoints of a diameter of the circle (x − 4)² + (y − 4)² = 18.

    You must show two things:

    • (i) The midpoint of AB equals the centre of the circle.
    • (ii) The radius of the circle equals half the length of AB.

  10. Tangent to a circle. Problem Solving

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

    • (a) Find the gradient of the radius from the centre (0, 0) to the point (3, 4) on x² + y² = 25.
    • (b) Hence find the equation of the tangent to x² + y² = 25 at (3, 4). Write your answer in the form ax + by = c.
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