[discia] GCSE Maths › Algebra › Equation of a circle

Equation of a circle

discia.org 10 questions · 37 marks
Name Date
  1. Either paper 2 marks
    (a)

    A circle has centre \(O\), the origin, and radius \(18\).

    Write down the equation of the circle.

    (b)

    A different circle has equation \(x^2 + y^2 = 196\)

    Write down the radius of this circle.

    Show answer
    (a) \(x^2 + y^2 = 324\)
    (b) \(14\)
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  2. Either paper 3 marks

    The circle \(C\) has equation \(x^2 + y^2 = 9\)

    (a)

    Write down the coordinates of the two points where \(C\) crosses the \(y\)-axis.

    (b)

    State the coordinates of the point where \(C\) crosses the negative \(x\)-axis.

    Show answer
    (a) \((0, 3)\) and \((0, -3)\)
    (b) \((-3, 0)\)
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  3. Calculator 3 marks

    A circle has centre the origin and equation \(x^2 + y^2 = 52\)

    (a)

    Calculate the radius of the circle.

    Give your answer in the form \(a\sqrt{b}\), where \(a\) and \(b\) are integers and \(b\) is as small as possible.

    (b)

    Calculate the radius correct to 1 decimal place.

    Show answer
    (a) \(2\sqrt{ 13 }\)
    (b) \(7.2\)
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  4. Either paper 3 marks

    The circle \(C\) has equation \(x^2 + y^2 = 400\)

    Does the point \((-12, 17)\) lie inside \(C\), on \(C\), or outside \(C\)?

    You must show your working.

    Show answer
    Outside the circle
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  5. Either paper 3 marks

    A circle has centre \(O\), the origin, and passes through the point \((-12, 5)\).

    (a)

    Find the equation of the circle.

    (b)

    Write down the radius of the circle.

    Show answer
    (a) \(x^2 + y^2 = 169\)
    (b) \(13\)
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  6. Either paper 7 marks

    \(A\), \(B\) and \(C\) are points on the circle \(x^2 + y^2 = 64\). The circle is not drawn; the points are shown on the axes.

    \(A\) is on the \(y\)-axis.

    \(B\) is on the \(x\)-axis.

    \(M\) is the midpoint of \(AB\).

    \(COM\) is a straight line.

    xyOABCM
    (a)

    Show that the coordinates of \(A\) are \((0, 8)\).

    (b)

    Work out the coordinates of \(B\).

    (c)

    Show that the equation of the straight line passing through \(C\), \(O\) and \(M\) is \(y = -x\).

    (d)

    Work out the coordinates of \(C\).

    Give your answer in surd form.

    Show answer
    (a) \(y^2 = 64\), so \(y = 8\) (above \(O\)): \(A = (0, 8)\)
    (b) \((-8, 0)\)
    (c) \(M = (-4, 4)\), gradient \(-1\), through \(O\): \(y = -x\)
    (d) \((4\sqrt{2}, -4\sqrt{2})\)
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  7. Either paper 5 marks
    xy-10-8-6-4-2246810-10-9-8-7-6-5-4-3-2123456789100
    (a)

    On the grid, draw the graph of \(x^2 + y^2 = 64\)

    On the same grid, draw the graph of \(6y = -x\)

    (b)

    Use your graphs to find estimates for the solutions of the simultaneous equations

    \(x^2 + y^2 = 64\)

    \(6y = -x\)

    Show answer
    (a) A circle with centre \(O\) (the origin) and radius \(8\), and the straight line through \(O\) and \((6, -1)\)
    (b) About \((7.9, -1.3)\) and \((-7.9, 1.3)\) (each coordinate within 0.3)
    (a)xy-10-8-6-4-2246810-10-9-8-7-6-5-4-3-2123456789100
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  8. Either paper 3 marks

    The point \((15, k)\) lies on the circle \(x^2 + y^2 = 625\)

    Given that \(k\) is positive, work out the value of \(k\).

    Show answer
    \(20\)
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  9. Calculator 4 marks

    The diagram shows a circle, centre \(O\), drawn on coordinate axes with \(O\) at the origin.

    \(BA\) is the tangent to the circle at the point \(A\).

    Angle \(OBA = 52^\circ\)

    Point \(B\) has coordinates \((27, 0)\).

    ACBO

    Point \(P\) has coordinates \((2p, p)\) and lies on the circle, where \(p > 0\).

    Work out the value of \(p\).

    Give your answer correct to 1 decimal place.

    You must show all your working.

    Show answer
    \(9.5\)
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  10. Either paper 4 marks

    A circle has equation \(x^2 + y^2 = 100\)

    The point \(P(-6, 8)\) lies on the circle.

    Nadia says that the tangent to the circle at \(P\) crosses the \(y\)-axis at the point \((0, 16)\).

    Is Nadia correct?

    You must show how you get your answer.

    Show answer
    No: the tangent crosses the \(y\)-axis at \((0, 12.5)\)
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discia.org Equation of a circle · sheet 1
Answers Equation of a circle · sheet 1
  1. (a) \(x^2 + y^2 = 324\)
    (b) \(14\)
  2. (a) \((0, 3)\) and \((0, -3)\)
    (b) \((-3, 0)\)
  3. (a) \(2\sqrt{ 13 }\)
    (b) \(7.2\)
  4. Outside the circle
  5. (a) \(x^2 + y^2 = 169\)
    (b) \(13\)
  6. (a) \(y^2 = 64\), so \(y = 8\) (above \(O\)): \(A = (0, 8)\)
    (b) \((-8, 0)\)
    (c) \(M = (-4, 4)\), gradient \(-1\), through \(O\): \(y = -x\)
    (d) \((4\sqrt{2}, -4\sqrt{2})\)
  7. (a) A circle with centre \(O\) (the origin) and radius \(8\), and the straight line through \(O\) and \((6, -1)\)
    (b) About \((7.9, -1.3)\) and \((-7.9, 1.3)\) (each coordinate within 0.3)
    (a)xy-10-8-6-4-2246810-10-9-8-7-6-5-4-3-2123456789100
  8. \(20\)
  9. \(9.5\)
  10. No: the tangent crosses the \(y\)-axis at \((0, 12.5)\)
discia.org Equation of a circle · sheet 1