Skip to content

Form 5 Equations of Circles Practice

Form 5 · 40 practice questions with instant marking and worked solutions.

Equations of Circles is an important part of the Form 5 mathematics curriculum. This topic develops centre, radius, standard equation and geometric relationships. Begin by identifying what every number, symbol, diagram and unit represents. Translate the information into a mathematical relationship before calculating. Each line of working should have a purpose. Finally use estimation, substitution, an inverse operation, or a second representation to check the result. This habit separates a misunderstanding from a copying error and makes correction more useful.

A reliable approach to Equations of Circles has four stages. First, mark the given information and the quantity required. Second, choose an appropriate representation, property or formula. Third, carry out the operations in a clear order while retaining signs and units. Fourth, return to the original context and decide whether the result is possible. For a graph or table, read the title, scale and labels before using a value. For algebra, inspect brackets, negative signs and the order of operations before simplifying.

The forty original questions in this topic use a shared mathematical specification in Chinese and English. Difficulty rises from level 1 to level 5. Early questions establish the language and direct skills of Equations of Circles; middle questions add two-step reasoning, conversions and common variations; later questions connect ideas and require a deliberate check. In multiple-choice work, calculate before looking at the choices. In fill-in and short-answer work, retain decisive steps. For true-or-false statements, support the decision with a calculation or counterexample. Four of them are DSE Paper 1 style structured questions answered in parts (a), (b), (c), where later parts build on earlier ones.

Effective revision is more than recording a score. After completing the Equations of Circles set, group errors by concept tags and decide whether each problem arose from interpretation, forming an expression, calculation, or presenting the answer. Retry the item on a later day without opening the solution, then test the same method with a different value. If the same tag appears repeatedly, return to the explanation and rebuild the concept. If signs or units are the issue, use a fixed pre-submission checklist.

Equations of Circles should also connect with the rest of mathematics. Number and algebra supply calculation tools; measurement and geometry give spatial meaning; data handling asks what a result means. A learner who can explain aloud why every step is valid has moved beyond imitation. Estimate a reasonable range, complete an exact solution, and compare the two. Agreement between context, estimate and exact answer is strong evidence that the solution is sound.

Start practising

  1. 1 Basic
    Find the radius of $(x-h)^2+(y-k)^2=16$. Choose the correct answer.
  2. 2 Basic
    Find the radius of $(x-h)^2+(y-k)^2=25$. Give the numerical value only.

  3. 3 Basic
    Find the radius of $(x-h)^2+(y-k)^2=36$. Show the essential working and give the numerical value.

  4. 4 Basic
    The radius of $(x-h)^2+(y-k)^2=9$ is 4. Is this true?
  5. 5 Basic
    Circle $C$ has equation $x^2+y^2+8x-4y-16=0$. Find the radius of $C$. Choose the correct answer.
  6. 6 Basic
    Circle $C$ has equation $x^2+y^2-10x+4y+4=0$. Find the $x$-coordinate of its centre. Give the answer only.

  7. 7 Basic
    A circle has centre $(-3,-5)$ and radius 2; its equation is written $x^2+y^2+Dx+Ey+F=0$. Find the value of $F$. Show the essential working and give the answer.

  8. 8 Basic
    Consider the points $A(5,1)$ and $B(-7,-15)$. The length of $AB$ is 28. Is this true?
  9. 9 Standard
    Find the radius of $(x-h)^2+(y-k)^2=25$. Choose the correct answer.
  10. 10 Standard
    Find the radius of $(x-h)^2+(y-k)^2=36$. Give the numerical value only.

  11. 11 Standard
    Find the radius of $(x-h)^2+(y-k)^2=49$. Show the essential working and give the numerical value.

  12. 12 Standard
    The radius of $(x-h)^2+(y-k)^2=16$ is 4. Is this true?
  13. 13 Standard
    A moving point $P$ stays 3 units from $C(-3,5)$; its locus is $x^2+y^2+6x-10y+F=0$. Find the value of $F$. Choose the correct answer.
  14. 14 Standard
    Circle $C$ has equation $x^2+y^2-12x-8y+43=0$. Find the radius of $C$. Give the answer only.

  15. 15 Standard
    Circle $C$ has equation $x^2+y^2+14x-10y+70=0$. Find the $x$-coordinate of its centre. Show the essential working and give the answer.

  16. 16 Standard DSE long question
    Circle $C$ has equation $x^2+y^2+6x+2y-279=0$. Answer each part; give the answer only.
    1. (a) Find the $x$-coordinate of the centre.
    2. (b) Find the $y$-coordinate of the centre.
    3. (c) Find the radius.
    4. (d) Find the distance from the point $P(5,14)$ to the centre of $C$.
  17. 17 Advanced
    Find the radius of $(x-h)^2+(y-k)^2=36$. Choose the correct answer.
  18. 18 Advanced
    Find the radius of $(x-h)^2+(y-k)^2=49$. Give the numerical value only.

  19. 19 Advanced
    Find the radius of $(x-h)^2+(y-k)^2=64$. Show the essential working and give the numerical value.

  20. 20 Advanced
    The radius of $(x-h)^2+(y-k)^2=25$ is 6. Is this true?
  21. 21 Advanced
    A circle has centre $(6,-4)$ and radius 4; its equation is written $x^2+y^2+Dx+Ey+F=0$. Find the value of $F$. Choose the correct answer.
  22. 22 Advanced
    Consider the points $A(-1,-4)$ and $B(20,16)$. Find the length of $AB$. Give the answer only.

  23. 23 Advanced
    A moving point $P$ stays 7 units from $C(-4,-4)$; its locus is $x^2+y^2+8x+8y+F=0$. The value of $F$ is -17. Is this true?
  24. 24 Advanced DSE long question
    $A(-4,-4)$ and $B(8,12)$. Answer each part; give the answer only.
    1. (a) Find the gradient of $AB$ (as a fraction in lowest terms).
    2. (b) Find the length of $AB$.
    3. (c) Find the $y$-intercept of the perpendicular bisector of $AB$ (as a fraction in lowest terms).
  25. 25 Challenging
    Find the radius of $(x-h)^2+(y-k)^2=49$. Choose the correct answer.
  26. 26 Challenging
    Find the radius of $(x-h)^2+(y-k)^2=64$. Give the numerical value only.

  27. 27 Challenging
    Find the radius of $(x-h)^2+(y-k)^2=81$. Show the essential working and give the numerical value.

  28. 28 Challenging
    The radius of $(x-h)^2+(y-k)^2=36$ is 6. Is this true?
  29. 29 Challenging
    Circle $C$ has equation $x^2+y^2-10x-6y+9=0$. Find the radius of $C$. Choose the correct answer.
  30. 30 Challenging
    Circle $C$ has equation $x^2+y^2-8x+2y-8=0$. Find the $y$-coordinate of its centre. Show the essential working and give the answer.

  31. 31 Challenging
    A circle has centre $(1,-1)$ and radius 5; its equation is written $x^2+y^2+Dx+Ey+F=0$. The value of $F$ is 27. Is this true?
  32. 32 Challenging DSE long question
    Circle $C$ has equation $x^2+y^2+10x-8y-128=0$. Answer each part; give the answer only.
    1. (a) Find the $x$-coordinate of the centre.
    2. (b) Find the $y$-coordinate of the centre.
    3. (c) Find the radius.
    4. (d) Find the distance from the point $P(0,16)$ to the centre of $C$.
  33. 33 Elite
    Find the radius of $(x-h)^2+(y-k)^2=64$. Choose the correct answer.
  34. 34 Elite
    Find the radius of $(x-h)^2+(y-k)^2=81$. Give the numerical value only.

  35. 35 Elite
    Find the radius of $(x-h)^2+(y-k)^2=100$. Show the essential working and give the numerical value.

  36. 36 Elite
    The radius of $(x-h)^2+(y-k)^2=49$ is 8. Is this true?
  37. 37 Elite
    Consider the points $A(-5,2)$ and $B(15,23)$. Find the length of $AB$. Give the answer only.

  38. 38 Elite
    A moving point $P$ stays 2 units from $C(5,-2)$; its locus is $x^2+y^2-10x+4y+F=0$. Find the value of $F$. Show the essential working and give the answer.

  39. 39 Elite
    Circle $C$ has equation $x^2+y^2+6x+6y-63=0$. The radius of $C$ is 9. Is this true?
  40. 40 Elite DSE long question
    $A(2,-2)$ and $B(20,22)$. Answer each part; give the answer only.
    1. (a) Find the gradient of $AB$ (as a fraction in lowest terms).
    2. (b) Find the length of $AB$.
    3. (c) Find the $y$-intercept of the perpendicular bisector of $AB$ (as a fraction in lowest terms).

The report lists what you got wrong and why. Only you can see it.

Common mistakes

Common mistakes in Equations of Circles

This topic needs particular care with centre, radius, standard equation and geometric relationships. The first mistake is applying a remembered rule before reading all conditions. The second is compressing working so much that a sign, bracket, decimal point or unit disappears. The third is accepting an impossible answer without checking: for example, a probability above one, a negative physical length, an angle outside the geometric constraints, or a result far from the estimate. Separate the conditions, model, calculation and check, and finish by stating what the answer represents.

More Form 5 topics

Form 5 mixed revision → My mistakes →

Keep missing the same type?

Repeating the same mistake usually means an earlier concept never landed. We find that step in the trial lesson.

Book a Trial WhatsApp us
WhatsApp