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Form 4 Loci Practice

Form 4 · 60 practice questions with instant marking and worked solutions.

Loci is an important part of the Form 4 mathematics curriculum. This topic develops moving-point conditions, equidistant sets, construction and intersections. 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 Loci 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 60 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 Loci; 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. 8 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 Loci 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.

Loci 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.

This batch adds 20 questions, bringing the topic to 60. The new questions cover controlled variations of different concepts, with paired bilingual conditions, hints, worked solutions and reproducible checks.

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  1. 1 Basic
    A and B have x-coordinates 4 and 10. At what x-coordinate does their equidistant locus meet the x-axis? Choose the correct answer.
  2. 2 Basic
    A and B have x-coordinates 6 and 12. At what x-coordinate does their equidistant locus meet the x-axis? Give the numerical value only.

  3. 3 Basic
    A and B have x-coordinates 8 and 14. At what x-coordinate does their equidistant locus meet the x-axis? Show the essential working and give the numerical value.

    Reproducible diagram for Loci; values and labels are stated in the question

  4. 4 Basic
    For A and B at x=10 and x=16, the equidistant locus meets the x-axis at 14. Is this true?
  5. 5 Basic
    A moving point $P$ is equidistant from $A(0,-1)$ and $B(4,1)$. Find the $y$-intercept of the locus of $P$ (a straight line). Choose the correct answer.
  6. 6 Basic
    A moving point $P$ stays 5 units from $C(-1,-5)$; its locus is $x^2+y^2+2x+10y+F=0$. Find the value of $F$. Give the answer only.

  7. 7 Basic
    A moving point $P$ is equidistant from the lines $y=4$ and $y=8$. Find the value of $c$ in its locus $y=c$. Show the essential working and give the answer.

  8. 8 Basic
    Consider the points $A(0,2)$ and $B(-9,-10)$. The length of $AB$ is 21. Is this true?
  9. 9 Basic
    A moving point $P$ is equidistant from $A(-4,2)$ and $B(0,4)$. Find the $y$-intercept of the locus of $P$ (a straight line). Choose the correct answer.
  10. 10 Basic
    A moving point $P$ stays 2 units from $C(5,-1)$; its locus is $x^2+y^2-10x+2y+F=0$. Find the value of $F$. Give the answer only.

  11. 11 Basic
    A moving point $P$ is equidistant from the lines $y=-2$ and $y=4$. Find the value of $c$ in its locus $y=c$. Show the essential working and give the answer.

  12. 12 Basic
    Consider the points $A(-1,-4)$ and $B(6,20)$. The length of $AB$ is 31. Is this true?
  13. 13 Standard
    A and B have x-coordinates 2 and 10. At what x-coordinate does their equidistant locus meet the x-axis? Choose the correct answer.
  14. 14 Standard
    A and B have x-coordinates 4 and 12. At what x-coordinate does their equidistant locus meet the x-axis? Give the numerical value only.

  15. 15 Standard
    A and B have x-coordinates 6 and 14. At what x-coordinate does their equidistant locus meet the x-axis? Show the essential working and give the numerical value.

  16. 16 Standard
    For A and B at x=8 and x=16, the equidistant locus meets the x-axis at 12. Is this true?
  17. 17 Standard
    A circle has centre $(3,6)$ and radius 2; its equation is written $x^2+y^2+Dx+Ey+F=0$. Find the value of $F$. Choose the correct answer.
  18. 18 Standard
    A moving point $P$ is equidistant from $A(-2,2)$ and $B(2,8)$. Find the $y$-intercept of the locus of $P$ (a straight line). Give the answer only.

  19. 19 Standard
    A moving point $P$ stays 4 units from $C(-4,5)$; its locus is $x^2+y^2+8x-10y+F=0$. Find the value of $F$. Show the essential working and give the answer.

  20. 20 Standard DSE long question
    $A(-1,-1)$ and $B(3,5)$; a moving point $P$ satisfies $PA=PB$. Answer each part; give the answer only.
    1. (a) Find the $y$-coordinate of the midpoint of $AB$.
    2. (b) Find the gradient of $AB$ (as a fraction in lowest terms).
    3. (c) Find the gradient of the locus of $P$ (as a fraction in lowest terms).
    4. (d) Find the $y$-intercept of the locus (as a fraction in lowest terms).
  21. 21 Standard
    A circle has centre $(5,4)$ 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 Standard
    A moving point $P$ is equidistant from $A(3,0)$ and $B(9,-2)$. Find the $y$-intercept of the locus of $P$ (a straight line). Give the answer only.

  23. 23 Standard
    A moving point $P$ stays 2 units from $C(-1,-4)$; its locus is $x^2+y^2+2x+8y+F=0$. Find the value of $F$. Show the essential working and give the answer.

  24. 24 Standard DSE long question
    $A(-3,-3)$ and $B(-1,3)$; a moving point $P$ satisfies $PA=PB$. Answer each part; give the answer only.
    1. (a) Find the $y$-coordinate of the midpoint of $AB$.
    2. (b) Find the gradient of $AB$.
    3. (c) Find the gradient of the locus of $P$ (as a fraction in lowest terms).
    4. (d) Find the $y$-intercept of the locus (as a fraction in lowest terms).
  25. 25 Advanced
    A and B have x-coordinates 10 and 20. At what x-coordinate does their equidistant locus meet the x-axis? Choose the correct answer.
  26. 26 Advanced
    A and B have x-coordinates 2 and 12. At what x-coordinate does their equidistant locus meet the x-axis? Give the numerical value only.

  27. 27 Advanced
    A and B have x-coordinates 4 and 14. At what x-coordinate does their equidistant locus meet the x-axis? Show the essential working and give the numerical value.

  28. 28 Advanced
    For A and B at x=6 and x=16, the equidistant locus meets the x-axis at 12. Is this true?
  29. 29 Advanced
    A moving point $P$ is equidistant from the lines $y=-2$ and $y=9$. Find the value of $c$ in its locus $y=c$ (as a fraction in lowest terms). Choose the correct answer.
  30. 30 Advanced
    Consider the points $A(-1,5)$ and $B(-17,-7)$. Find the length of $AB$. Give the answer only.

  31. 31 Advanced
    A circle has centre $(5,-1)$ and radius 6; its equation is written $x^2+y^2+Dx+Ey+F=0$. The value of $F$ is -10. Is this true?
  32. 32 Advanced DSE long question
    Circle $C$ has equation $x^2+y^2-4x-10y-140=0$. 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 $P(7,17)$ to the centre of $C$.
  33. 33 Advanced
    A moving point $P$ is equidistant from the lines $y=4$ and $y=16$. Find the value of $c$ in its locus $y=c$. Choose the correct answer.
  34. 34 Advanced
    Consider the points $A(-1,-5)$ and $B(-4,-1)$. Find the length of $AB$. Give the answer only.

  35. 35 Advanced
    A circle has centre $(-3,4)$ and radius 7; its equation is written $x^2+y^2+Dx+Ey+F=0$. The value of $F$ is -24. Is this true?
  36. 36 Advanced DSE long question
    Circle $C$ has equation $x^2+y^2+4x-4y-281=0$. 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 $P(6,17)$ to the centre of $C$.
  37. 37 Challenging
    A and B have x-coordinates 8 and 20. At what x-coordinate does their equidistant locus meet the x-axis? Choose the correct answer.
  38. 38 Challenging
    A and B have x-coordinates 10 and 22. At what x-coordinate does their equidistant locus meet the x-axis? Give the numerical value only.

  39. 39 Challenging
    A and B have x-coordinates 2 and 14. At what x-coordinate does their equidistant locus meet the x-axis? Show the essential working and give the numerical value.

  40. 40 Challenging
    For A and B at x=4 and x=16, the equidistant locus meets the x-axis at 10. Is this true?
  41. 41 Challenging
    A moving point $P$ is equidistant from $A(-4,3)$ and $B(0,5)$. Find the $y$-intercept of the locus of $P$ (a straight line). Choose the correct answer.
  42. 42 Challenging
    A moving point $P$ stays 4 units from $C(-4,3)$; its locus is $x^2+y^2+8x-6y+F=0$. Find the value of $F$. Show the essential working and give the answer.

  43. 43 Challenging
    A moving point $P$ is equidistant from the lines $y=4$ and $y=7$. The value of $c$ in its locus $y=c$ is 3. Is this true?
  44. 44 Challenging DSE long question
    $A(2,-3)$ and $B(8,-1)$; a moving point $P$ satisfies $PA=PB$. Answer each part; give the answer only.
    1. (a) Find the $y$-coordinate of the midpoint of $AB$.
    2. (b) Find the gradient of $AB$ (as a fraction in lowest terms).
    3. (c) Find the gradient of the locus of $P$.
    4. (d) Find the $y$-intercept of the locus.
  45. 45 Challenging
    A moving point $P$ is equidistant from $A(-2,4)$ and $B(0,0)$. Find the $y$-intercept of the locus of $P$ (a straight line) (as a fraction in lowest terms). Choose the correct answer.
  46. 46 Challenging
    A moving point $P$ stays 3 units from $C(5,-3)$; its locus is $x^2+y^2-10x+6y+F=0$. Find the value of $F$. Show the essential working and give the answer.

  47. 47 Challenging
    A moving point $P$ is equidistant from the lines $y=-3$ and $y=9$. The value of $c$ in its locus $y=c$ is 12. Is this true?
  48. 48 Challenging DSE long question
    $A(-3,2)$ and $B(-1,6)$; a moving point $P$ satisfies $PA=PB$. Answer each part; give the answer only.
    1. (a) Find the $y$-coordinate of the midpoint of $AB$.
    2. (b) Find the gradient of $AB$.
    3. (c) Find the gradient of the locus of $P$ (as a fraction in lowest terms).
    4. (d) Find the $y$-intercept of the locus.
  49. 49 Elite
    A and B have x-coordinates 6 and 20. At what x-coordinate does their equidistant locus meet the x-axis? Choose the correct answer.
  50. 50 Elite
    A and B have x-coordinates 8 and 22. At what x-coordinate does their equidistant locus meet the x-axis? Give the numerical value only.

  51. 51 Elite
    A and B have x-coordinates 10 and 24. At what x-coordinate does their equidistant locus meet the x-axis? Show the essential working and give the numerical value.

  52. 52 Elite
    For A and B at x=2 and x=16, the equidistant locus meets the x-axis at 10. Is this true?
  53. 53 Elite
    Consider the points $A(-5,-4)$ and $B(3,-19)$. Find the length of $AB$. Give the answer only.

  54. 54 Elite
    A circle has centre $(2,5)$ 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.

  55. 55 Elite
    A moving point $P$ is equidistant from $A(-5,1)$ and $B(-1,7)$. The $y$-intercept of the locus of $P$ (a straight line) is 2. Is this true?
  56. 56 Elite DSE long question
    Circle $C$ has equation $x^2+y^2-2y-288=0$. 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 $P(8,16)$ to the centre of $C$.
  57. 57 Elite
    Consider the points $A(3,-1)$ and $B(18,7)$. Find the length of $AB$. Give the answer only.

  58. 58 Elite
    A circle has centre $(6,-5)$ 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.

  59. 59 Elite
    A moving point $P$ is equidistant from $A(3,-3)$ and $B(7,3)$. The $y$-intercept of the locus of $P$ (a straight line) is 10/3. Is this true?
  60. 60 Elite DSE long question
    Circle $C$ has equation $x^2+y^2-10x+8y+16=0$. 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 $P(8,0)$ to the centre of $C$.

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

Common mistakes

Common mistakes in Loci

This topic needs particular care with moving-point conditions, equidistant sets, construction and intersections. 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.

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