Practice Maths

L45 — Circles and Coordinate Geometry

Key Terms

Centre-radius form
(x − h)² + (y − k)² = r² — the standard equation of a circle with centre (h, k) and radius r.
Radius
The fixed distance from the centre to any point on the circle; found as r = √(right-hand side).
Tangent
A line that touches the circle at exactly one point and is perpendicular to the radius at that point.
Chord
A line segment joining two points on the circle.
Perpendicular bisector of a chord
The line that is both perpendicular to the chord and passes through its midpoint; it always passes through the centre of the circle.
Completing the square
An algebraic technique used to rewrite the general form x² + y² + Dx + Ey + F = 0 into centre-radius form.

Circle equations

FormEquationDetails
Centre-radius (x − h)² + (y − k)² = r² Centre (h, k), radius r
Origin-centred x² + y² = r² Centre (0, 0), radius r
General form x² + y² + Dx + Ey + F = 0 Complete the square to find centre and radius

Lines and circles

RelationshipCondition
Tangent to circle at point PPerpendicular to radius at P
ChordLine segment joining two points on the circle
Perpendicular bisector of chordPasses through the centre
Point inside / on / outside circledistance from centre < / = / > r
x y 1 2 3 4 5 6 0 1 2 3 4 5 (3, 2) T(3, 5) r = 3 tangent y = 5
Circle (x−3)²+(y−2)²=9, centre (3,2), radius 3. The tangent at T(3,5) is horizontal (y=5) — perpendicular to the vertical radius.
Hot Tip: To find if a point is inside, on, or outside a circle: compute (x−h)²+(y−k)² and compare to r². If the result is less than r² → inside; equal → on; greater → outside. No square roots needed.

Worked Example 1 — Equation of a circle

Write the equation of a circle with centre (2, −3) and radius 5.

(x − 2)² + (y + 3)² = 25.

Worked Example 2 — Reading circle properties

State the centre and radius of (x + 1)² + (y − 4)² = 36.

Compare with (x − h)² + (y − k)² = r²: h = −1, k = 4, r² = 36 ⇒ r = 6.

Centre: (−1, 4). Radius: 6.

Worked Example 3 — Is a point on/inside/outside the circle?

Circle: x² + y² = 25. Test points A(3, 4), B(4, 4), C(0, 4).

A: 3²+4² = 9+16 = 25 = r² ⇒ on the circle.

B: 4²+4² = 32 > 25 ⇒ outside.

C: 0+16 = 16 < 25 ⇒ inside.

Worked Example 4 — Tangent to a circle

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

Radius OP has gradient 4/3. Tangent is perpendicular: m = −3/4.

y − 4 = −3/4(x − 3) ⇒ y = −3x/4 + 9/4 + 4 = −3x/4 + 25/4.

Or: 3x + 4y = 25.

Worked Example 5 — Completing the square

Find the centre and radius of x² + y² − 6x + 4y − 12 = 0.

Group and complete the square:

(x² − 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4

(x − 3)² + (y + 2)² = 25.

Centre: (3, −2). Radius: 5.

See Answers ➔

  1. Equation of a circle. Fluency

    • (a) Write the equation of a circle with centre (0, 0) and radius 7.
    • (b) Write the equation of a circle with centre (4, −1) and radius 3.
    • (c) State the centre and radius of (x − 5)² + (y + 2)² = 16.
    • (d) State the centre and radius of x² + y² = 49.

  2. Points and circles. Fluency

    • (a) Is the point (6, 8) on, inside or outside the circle x² + y² = 100?
    • (b) Is the point (3, 4) on, inside or outside the circle (x − 1)² + (y − 1)² = 25?
    • (c) Find four points on the circle x² + y² = 25 that lie on the axes.
    • (d) A circle has centre (0, 0). The point (5, 12) is on the circle. Write its equation.

  3. Tangents to circles. Fluency

    • (a) Find the gradient of the radius to the point (0, 5) on the circle x² + y² = 25. Hence find the tangent equation at that point.
    • (b) Find the equation of the tangent to x² + y² = 25 at the point (4, −3).
    • (c) Find the equation of the tangent to (x − 2)² + (y − 1)² = 10 at the point (5, 2).
    • (d) A tangent to the circle x² + y² = 50 at the point (5, 5). Find the equation of this tangent.

  4. Completing the square. Fluency

    • (a) Find the centre and radius of x² + y² − 4x + 6y + 4 = 0.
    • (b) Find the centre and radius of x² + y² + 8x − 2y − 8 = 0.
    • (c) Write x² + y² − 10x = 0 in centre-radius form.
    • (d) Show that x² + y² − 2x − 4y + 5 = 0 represents a single point (degenerate circle). State the point.

  5. Chords and the centre. Understanding

    A circle has equation x² + y² = 25. The points A(3, 4) and B(−3, 4) are both on the circle.

    x y 3 −3 4 −4 0 O(0,0) A(3,4) B(−3,4) x = 0 M(0,4)
    Chord AB with perpendicular bisector passing through the centre O.
    • (a) Verify that A(3, 4) and B(−3, 4) both lie on x² + y² = 25.
    • (b) Find the midpoint M of chord AB and the gradient of AB.
    • (c) Write the equation of the perpendicular bisector of AB. Verify it passes through the centre (0, 0).
    • (d) Explain why the perpendicular bisector of any chord of a circle must pass through the centre.

  6. Intersections of circles and lines. Understanding

    • (a) Find where the line y = x + 1 intersects the circle x² + y² = 25.
    • (b) How many times does the line y = 4 intersect x² + y² = 9? Justify without solving.
    • (c) Show that the line y = 3x − 10 is a tangent to the circle x² + y² = 10.
    • (d) Find the length of the chord where y = x intersects (x − 2)² + y² = 8.

  7. Circle through given points. Understanding

    • (a) Find the centre of the circle passing through A(0, 0), B(6, 0) and C(0, 8). (Hint: use the perpendicular bisectors of two chords.)
    • (b) A circle has centre (2, 3) and passes through (5, 7). Write its equation.
    • (c) The endpoints of a diameter are P(1, 3) and Q(7, 3). Write the equation of the circle.
    • (d) A circle passes through (0, 0) and has centre (3, 4). Write its equation.

  8. Mixed coordinate geometry. Understanding

    • (a) Line y = 2 meets circle (x − 1)² + (y − 2)² = 9. Find the chord length.
    • (b) The tangent to x² + y² = r² at (a, b) has equation ax + by = r². Use this to find the tangent to x² + y² = 13 at (2, 3).
    • (c) Find the distance from the centre (3, 1) of the circle (x−3)²+(y−1)²=20 to the line 2x − y = 4. Compare with the radius.
    • (d) Two circles: x² + y² = 4 and (x−5)² + y² = 9. Do they overlap? Justify using the distance between centres.

  9. Circle in context. Problem Solving

    A circular running track has its centre at C(4, 3) on a coordinate grid where 1 unit = 100 m. The track passes through the point A(4, 8).

    • (a) Find the radius of the track in units, and hence the circumference in metres (to the nearest metre).
    • (b) Write the equation of the circular track.
    • (c) A straight path runs along y = 6. Find where this path intersects the track (give coordinates).
    • (d) A second path is tangent to the track at A(4, 8). Find the equation of this path.

  10. Coordinate geometry proof. Problem Solving

    The triangle formed by A(0, 0), B(10, 0), C(10, 10) is right-angled at B.

    • (a) Verify that the angle at B is a right angle using gradients of AB and BC.
    • (b) For a right-angled triangle, the hypotenuse is a diameter of the circumscribed circle. Find the centre and radius of the circumscribed circle of triangle ABC.
    • (c) Write the equation of the circumscribed circle and verify that all three vertices A, B, C lie on it.
    • (d) Find the equation of the tangent to the circumscribed circle at vertex A(0, 0).