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IGCSE Additional Mathematics (0606)

Coordinate geometry of the circle

3 questions · about 1% of the marks we indexed · examined in 3 of 9 sessions.

Questions by year

2026

  • 2026 May/June · Paper 22 · Q119 marksCircle and line (tangents, intersections)

2025

  • 2025 Oct/Nov · Paper 12 · Q7 (b)4 marksEquation of a circle
  • 2025 May/June · Paper 12 · Q3 (a, b, c)8 marksCircle and line (tangents, intersections) · Equation of a circle

References only, not question text. Open Official paper ↗ where Cambridge publishes the paper, or see the official Cambridge past-papers page; for other papers, ask your school. “Paper 42” means Paper 4, variant 2.

Practice questions

Original questions written by HomiLabs in the style of the exam, with answers and mark schemes. They are not Cambridge past paper questions and have not been reviewed by Cambridge. Try each one before opening the answer.

Q1Circle and line (tangents, intersections) · Core · Paper 1 style · no calculator · 6 marks
  1. (a) Find the coordinates of the points where the line y = x + 1 meets the circle x² + y² = 25. [4]
  2. (b) Find the exact length of the chord joining these points. [2]
Show answer
  1. (a) (3, 4) and (−4, −3)
    x² + (x + 1)² = 25 M1; x² + x − 12 = 0 A1; x = 3, −4 A1; points A1
  2. (b) 7√2
    √(7² + 7²) M1; = 7√2 A1
Q2Circle and line (tangents, intersections) · Core · Paper 1 style · no calculator · 3 marks
  1. (a) The point (3, 4) lies on the circle x² + y² = 25. Find the equation of the tangent to the circle at this point. [3]
Show answer
  1. (a) 3x + 4y = 25
    gradient of radius = 4/3 B1; tangent gradient −3/4 M1; y − 4 = −¾(x − 3) → 3x + 4y = 25 A1
Q3Circle and line (tangents, intersections) · Core · Paper 1 style · no calculator · 3 marks
  1. (a) Find the equation of the tangent to the circle (x − 2)² + (y + 1)² = 13 at the point (4, 2). Give your answer in the form ax + by = c. [3]
Show answer
  1. (a) 2x + 3y = 14
    radius gradient (2 + 1)/(4 − 2) = 3/2 B1; tangent gradient −2/3 M1; y − 2 = −⅔(x − 4) → 2x + 3y = 14 A1
Q4Circle and line (tangents, intersections) · Core · Paper 1 style · no calculator · 4 marks
  1. (a) Find the coordinates of the points where the circle (x − 3)² + (y + 4)² = 25 meets the coordinate axes. [4]
Show answer
  1. (a) (0, 0), (6, 0), (0, −8)
    y = 0: (x − 3)² = 9 → x = 0, 6 M1 A1; x = 0: (y + 4)² = 16 → y = 0, −8 M1 A1
Q5Circle and line (tangents, intersections) · Core · Paper 2 style · calculator · 3 marks
  1. (a) Determine whether the line y = 2x meets the circle x² + y² − 10x + 16 = 0. [3]
Show answer
  1. (a) No: the discriminant is −220 < 0
    x² + 4x² − 10x + 16 = 0 → 5x² − 10x + 16 = 0 M1; 100 − 320 = −220 M1; < 0 so the line does not meet the circle A1
Q6Circle and line (tangents, intersections) · Standard · Paper 1 style · no calculator · 5 marks
  1. (a) Show that the line y = 2x + 10 is a tangent to the circle (x − 1)² + (y − 2)² = 20, and find the coordinates of the point of contact. [5]
Show answer
  1. (a) Shown; point of contact (−3, 4)
    (x − 1)² + (2x + 8)² = 20 M1; 5x² + 30x + 45 = 0 → x² + 6x + 9 = 0 A1; discriminant 36 − 36 = 0 so tangent A1; x = −3 M1; y = 4 A1
Q7Circle and line (tangents, intersections) · Standard · Paper 1 style · no calculator · 4 marks
  1. (a) Find the values of c for which the line y = x + c is a tangent to the circle x² + y² = 8. [4]
Show answer
  1. (a) c = 4 or c = −4
    x² + (x + c)² = 8 → 2x² + 2cx + c² − 8 = 0 M1; 4c² − 8(c² − 8) = 0 M1; c² = 16 A1; c = ±4 A1
Q8Circle and line (tangents, intersections) · Standard · Paper 1 style · no calculator · 7 marks

The circle C has equation (x − 1)² + (y − 2)² = 25. The line y = 14 − 2x meets C at the points A and B, where A has the smaller x-coordinate.

  1. (a) Find the coordinates of A and B. [4]
  2. (b) Find the equation of the perpendicular bisector of AB and show that it passes through the centre of C. [3]
Show answer
  1. (a) A(4, 6), B(6, 2)
    (x − 1)² + (12 − 2x)² = 25 M1; 5x² − 50x + 120 = 0 → x² − 10x + 24 = 0 A1; x = 4, 6 A1; y = 6, 2 A1
  2. (b) y = ½x + 3/2; at x = 1, y = 2 (the centre)
    midpoint (5, 4), gradient ½ M1; y − 4 = ½(x − 5) A1; substitute (1, 2) A1
Q9Circle and line (tangents, intersections) · Standard · Paper 2 style · calculator · 4 marks
  1. (a) Show that the line 3x − 4y = 30 does not meet the circle x² + y² = 25. [4]
Show answer
  1. (a) Shown: discriminant −17600 < 0
    y = (3x − 30)/4 M1; 16x² + (3x − 30)² = 400 → 25x² − 180x + 500 = 0 A1; 180² − 4 × 25 × 500 = −17600 M1; < 0 so no intersection A1
Q10Circle and line (tangents, intersections) · Standard · Paper 2 style · calculator · 5 marks

A circle has centre (1, 1) and radius 4. The point P has coordinates (9, 7).

  1. (a) Find the shortest and the longest distances from P to points on the circle. [3]
  2. (b) A tangent from P touches the circle at T. Find the exact length PT. [2]
Show answer
  1. (a) shortest 6, longest 14
    distance from P to centre = √(64 + 36) = 10 M1 A1; 10 − 4 = 6, 10 + 4 = 14 A1
  2. (b) PT = 2√21
    PT² = 10² − 4² = 84 (radius ⟂ tangent) M1; PT = √84 = 2√21 A1
Q11Circle and line (tangents, intersections) · Standard · Paper 2 style · calculator · 7 marks

The circle x² + y² − 8x − 2y + 7 = 0 meets the line y = x − 1 at the points A and B.

  1. (a) Find the coordinates of A and B. [4]
  2. (b) Find the exact length of AB and the distance from the centre of the circle to the midpoint of AB. [3]
Show answer
  1. (a) (1, 0) and (5, 4)
    x² + (x − 1)² − 8x − 2(x − 1) + 7 = 0 M1; 2x² − 12x + 10 = 0 → x² − 6x + 5 = 0 A1; x = 1, 5 A1; y = 0, 4 A1
  2. (b) AB = 4√2; distance √2
    AB = √(16 + 16) = 4√2 B1; centre (4, 1), midpoint (3, 2) M1; √2 A1
Q12Circle and line (tangents, intersections) · Standard · Paper 1 style · no calculator · 4 marks

The point P(4, 3) lies on the circle (x + 1)² + (y − 2)² = 26.

  1. (a) Find the equation of the normal to the circle at P. [2]
  2. (b) Find the equation of the tangent to the circle at P. [2]
Show answer
  1. (a) x − 5y + 11 = 0
    the normal passes through the centre (−1, 2): gradient ⅕ M1; A1
  2. (b) 5x + y = 23
    gradient −5 M1; y − 3 = −5(x − 4) A1
Q13Circle and line (tangents, intersections) · Standard · Paper 2 style · calculator · 6 marks

On a map of a park in Islamabad, with 1 unit representing 10 m, the edge of a circular pond has equation x² + y² − 10x − 6y + 9 = 0. A straight path follows the line y = 2x − 12.

  1. (a) Find the coordinates of the points where the path meets the edge of the pond. [4]
  2. (b) Find the length of path that lies inside the pond, in metres, correct to 3 significant figures. [2]
Show answer
  1. (a) (5, −2) and (9, 6)
    substitute: 5x² − 70x + 225 = 0 M1 A1; x² − 14x + 45 = 0 → x = 5, 9 A1; y = −2, 6 A1
  2. (b) 89.4 m
    √(16 + 64) = √80 units M1; × 10 = 89.4 m A1
Q14Circle and line (tangents, intersections) · Challenge · Paper 1 style · no calculator · 5 marks
  1. (a) Find the gradients of the two tangents from the origin to the circle (x − 5)² + y² = 9. [5]
Show answer
  1. (a) m = ¾ and m = −¾
    line y = mx: (x − 5)² + m²x² = 9 M1; (1 + m²)x² − 10x + 16 = 0 A1; 100 − 64(1 + m²) = 0 M1; m² = 9/16 A1; m = ±¾ A1
Q15Circle and line (tangents, intersections) · Challenge · Paper 1 style · no calculator · 7 marks

The circle x² + y² − 4x + 6y − 12 = 0 passes through the point P(5, 1).

  1. (a) Find the equation of the tangent to the circle at P. [4]
  2. (b) The tangent meets the x-axis at Q and the y-axis at R. Find the exact area of triangle OQR, where O is the origin. [3]
Show answer
  1. (a) 3x + 4y = 19
    centre (2, −3) B1; radius gradient 4/3 M1; tangent gradient −3/4 M1; 3x + 4y = 19 A1
  2. (b) 361/24
    Q(19/3, 0), R(0, 19/4) B1 B1; ½ × 19/3 × 19/4 = 361/24 A1
Q16Circle and line (tangents, intersections) · Challenge · Paper 1 style · no calculator · 5 marks
  1. (a) Find the set of values of k for which the line y = kx + 5 meets the circle x² + y² = 5 at two distinct points. [5]
Show answer
  1. (a) k < −2 or k > 2
    x² + (kx + 5)² = 5 M1; (1 + k²)x² + 10kx + 20 = 0 A1; 100k² − 80(1 + k²) > 0 M1; k² > 4 A1; k < −2 or k > 2 A1
Q17Circle and line (tangents, intersections) · Challenge · Paper 1 style · no calculator · 8 marks

The points A(4, 2) and B(2, −4) lie on the circle x² + y² = 20, whose centre is O.

  1. (a) Find the equations of the tangents to the circle at A and at B. [4]
  2. (b) The tangents meet at T. Find the coordinates of T. [2]
  3. (c) Find the area of the quadrilateral OATB. [2]
Show answer
  1. (a) at A: 2x + y = 10; at B: x − 2y = 10
    radius OA gradient ½ → tangent gradient −2 M1; 2x + y = 10 A1; radius OB gradient −2 → tangent gradient ½ M1; x − 2y = 10 A1
  2. (b) T(6, −2)
    solve simultaneously M1 A1
  3. (c) 20
    two right-angled triangles, each ½ × √20 × √20 = 10 M1; total 20 A1
Q18Circle and line (tangents, intersections) · Challenge · Paper 1 style · no calculator · 6 marks
  1. (a) The line y = x + k is a tangent to the circle (x − 3)² + (y − 1)² = 8. Find the possible values of k and the coordinates of the point of contact in each case. [6]
Show answer
  1. (a) k = 2, contact (1, 3); k = −6, contact (5, −1)
    (x − 3)² + (x + k − 1)² = 8 → 2x² + (2k − 8)x + (k − 1)² + 1 = 0 M1 A1; (2k − 8)² − 8((k − 1)² + 1) = 0 → k² + 4k − 12 = 0 M1 A1; k = 2: x = 1, y = 3 A1; k = −6: x = 5, y = −1 A1
Q19Circle and line (tangents, intersections) · Challenge · Paper 1 style · no calculator · 6 marks
  1. (a) A circle has centre (2, −1) and touches the line 3x + 4y = 12. By finding the foot of the perpendicular from the centre to the line, find the radius and the equation of the circle. [6]
Show answer
  1. (a) radius 2; (x − 2)² + (y + 1)² = 4
    perpendicular gradient 4/3: y + 1 = (4/3)(x − 2) M1 A1; solve with 3x + 4y = 12: x = 16/5, y = 3/5 M1 A1; radius² = (6/5)² + (8/5)² = 4 M1; equation A1
Q20Circle and line (tangents, intersections) · Challenge · Paper 1 style · no calculator · 6 marks

Circle C₁ has equation x² + y² = 25 and circle C₂ has equation x² + y² − 8x − 6y + 15 = 0.

  1. (a) Show that the points of intersection of C₁ and C₂ lie on the line 4x + 3y = 20. [2]
  2. (b) Find the coordinates of the points of intersection of the two circles. [4]
Show answer
  1. (a) Shown
    subtract the equations: 8x + 6y − 15 = 25 M1; 4x + 3y = 20 A1
  2. (b) (5, 0) and (1.4, 4.8)
    y = (20 − 4x)/3 into C₁ M1; 25x² − 160x + 175 = 0 → 5x² − 32x + 35 = 0 A1; x = 5, 7/5 A1; y = 0, 24/5 A1
Q21Equation of a circle · Core · Paper 1 style · no calculator · 2 marks
  1. (a) Write down the equation of the circle with centre (3, −2) and radius 5. [1]
  2. (b) Show that the point (6, 2) lies on this circle. [1]
Show answer
  1. (a) (x − 3)² + (y + 2)² = 25
    B1
  2. (b) Shown
    3² + 4² = 25 B1
Q22Equation of a circle · Core · Paper 1 style · no calculator · 3 marks
  1. (a) Find the centre and radius of the circle x² + y² − 6x + 10y + 9 = 0. [3]
Show answer
  1. (a) centre (3, −5), radius 5
    (x − 3)² + (y + 5)² = 9 + 25 − 9 M1; centre (3, −5) A1; radius 5 A1
Q23Equation of a circle · Core · Paper 1 style · no calculator · 3 marks
  1. (a) A circle has centre (−4, 3) and touches the x-axis. Write down its equation, and explain why it does not touch the y-axis. [3]
Show answer
  1. (a) (x + 4)² + (y − 3)² = 9; the distance from the centre to the y-axis is 4, which is greater than the radius 3
    radius = 3 (distance to the x-axis) B1; equation B1; reason B1
Q24Equation of a circle · Standard · Paper 1 style · no calculator · 4 marks
  1. (a) The points A(−1, 4) and B(7, −2) are the ends of a diameter of a circle. Find the equation of the circle. [4]
Show answer
  1. (a) (x − 3)² + (y − 1)² = 25
    centre = midpoint (3, 1) M1 A1; AB = √(64 + 36) = 10 so radius 5 M1; equation A1
Q25Equation of a circle · Standard · Paper 1 style · no calculator · 3 marks
  1. (a) Find the centre and radius of the circle 3x² + 3y² − 12x + 18y − 9 = 0. [3]
Show answer
  1. (a) centre (2, −3), radius 4
    divide by 3: x² + y² − 4x + 6y − 3 = 0 M1; (x − 2)² + (y + 3)² = 16 A1; centre (2, −3), radius 4 A1
Q26Equation of a circle · Standard · Paper 1 style · no calculator · 5 marks
  1. (a) The centre of a circle lies on the line y = 2x. The circle passes through the points (5, 0) and (1, 4). Find the equation of the circle. [5]
Show answer
  1. (a) (x + 1)² + (y + 2)² = 40
    perpendicular bisector of the chord: midpoint (3, 2), chord gradient −1 so bisector y = x − 1 M1 A1; meet y = 2x: x = −1, centre (−1, −2) M1 A1; r² = 36 + 4 = 40 A1
Q27Equation of a circle · Standard · Paper 1 style · no calculator · 4 marks
  1. (a) Find the centre and radius of the circle x² + y² − 2x − 4y − 20 = 0. [2]
  2. (b) Determine whether the point P(4, 7) lies inside, on or outside the circle. [2]
Show answer
  1. (a) centre (1, 2), radius 5
    (x − 1)² + (y − 2)² = 25 M1 A1
  2. (b) outside, since the distance from the centre is √34 > 5
    distance² = 9 + 25 = 34 M1; 34 > 25 so outside A1
Q28Equation of a circle · Challenge · Paper 1 style · no calculator · 8 marks

A circle passes through the points A(−2, 3), B(4, 5) and C(6, −1).

  1. (a) Find the equation of the perpendicular bisector of AB. [3]
  2. (b) Find the equation of the perpendicular bisector of BC. [2]
  3. (c) Hence find the equation of the circle. [3]
Show answer
  1. (a) y = −3x + 7
    midpoint (1, 4) B1; gradient AB = ⅓ so perpendicular gradient −3 M1; y = −3x + 7 A1
  2. (b) y = ⅓x + ⅓ (or x − 3y + 1 = 0)
    midpoint (5, 2), gradient BC = −3 so perpendicular gradient ⅓ M1; A1
  3. (c) (x − 2)² + (y − 1)² = 20
    solve: centre (2, 1) M1 A1; r² = 16 + 4 = 20 A1
Q29Equation of a circle · Challenge · Paper 1 style · no calculator · 4 marks
  1. (a) The circle x² + y² − 4x + ky + 4 = 0 has radius 3. Find the possible values of the constant k, and state the centre in each case. [4]
Show answer
  1. (a) k = 6, centre (2, −3); k = −6, centre (2, 3)
    (x − 2)² + (y + k/2)² = 4 + k²/4 − 4 = k²/4 M1 A1; k²/4 = 9 → k = ±6 A1; centres (2, −k/2) A1
Q30Equation of a circle · Challenge · Paper 2 style · calculator · 5 marks

Circle C₁ has equation (x − 1)² + (y − 2)² = 9 and circle C₂ has equation (x − 7)² + (y − 10)² = 49.

  1. (a) Show that the circles touch each other. [3]
  2. (b) Find the coordinates of the point where they touch. [2]
Show answer
  1. (a) Shown: distance between centres = 10 = 3 + 7
    centres (1, 2), (7, 10); radii 3, 7 B1; distance √(36 + 64) = 10 M1; = sum of radii, so they touch (externally) A1
  2. (b) (2.8, 4.4)
    point is 3/10 of the way from (1, 2) to (7, 10) M1; (1 + 1.8, 2 + 2.4) A1

Similar questions

Questions that test the same subtopic in different years, grouped so you can work on one skill at a time.

References only: find each question in its paper (official link where available, otherwise through your school).

Circle and line (tangents, intersections) Rare2 questions · 2/9 sessions
Equation of a circle Rare2 questions · 2/9 sessions

Subtopics

Our analysis of how often each subtopic was examined and how many marks it carried, 2022 to May/June 2026.

SubtopicMarks indexedSessions
Circle and line (tangents, intersections) Rare142/9
Equation of a circle Rare72/9

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