UniGeniusGCSE Maths · Edexcel 1MA1

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Origin-centred circles and tangents

3 exam-style questions. Answers and the mark for each step are on the last page.

  1. Question 1Non-calculator · 3 marks

    A circle has equation x2+y2=49x^2 + y^2 = 49.

    (a) Write down the radius of the circle. (1)

    (b) Does the point (5,5)(5, 5) lie inside, on or outside the circle? You must show how you get your answer. (2)

    1. Inside
    2. On the circle
    3. Outside
  2. Question 2Non-calculator · 5 marks

    The circle has equation x2+y2=100x^2+y^2=100. Point P is (6,8)(6,8).

    (a) Show that P is on the circle. (2)

    (b) Find an equation of the tangent at P. (3)

  3. Question 3Non-calculator · 3 marks

    (a) The line y = 2x + c is tangent to the circle x2x^{2} + y2y^{2} = 20. Find both possible values of c. (3)

Answers and marks

Question 1

(a) 77

  • B1 The correct answer, 77.

(b) Outside: 52+52=50>495^2 + 5^2 = 50 > 49.

  • M1 Working out 52+52=505^2 + 5^2 = 50.
  • C1 Outside, because 50>4950 > 49.

Question 2

(a) 62+82=1006^2+8^2=100, matching the circle equation.

  • M1 Substitute both coordinates and add their squares.
  • C1 Correct conclusion with supporting reasoning: 62+82=1006^2+8^2=100, matching the circle equation.

(b) 6x+8y=1006x+8y=100

  • M1 The radius gradient is 4/3, so the tangent gradient is −3/4.
  • M1 Use P in the point-gradient equation.
  • A1 Correct answer: 6x+8y=1006x+8y=100

Question 3

(a) −10,10-10, 10

  • P1 Establishing 5x2+4cx+c2−20=05x^{2}+4cx+c^{2}-20=0 or an equivalent valid method.
  • P1 Establishing 400−4c2=0400-4c^{2}=0 or an equivalent valid method.
  • A1 Correct answer: −10,10-10, 10

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Origin-centred circles and tangents

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