1.03e Complete the square: find centre and radius of circle

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AQA Paper 3 2024 June Q9
9 marks Moderate -0.3
Figure 1 below shows a circle. **Figure 1** \includegraphics{figure_9} The centre of the circle is \(P\) and the circle intersects the \(y\)-axis at \(Q\) as shown in Figure 1. The equation of the circle is $$x^2 + y^2 = 12y - 8x - 27$$ \begin{enumerate}[label=(\alph*)] \item Express the equation of the circle in the form $$(x - a)^2 + (y - b)^2 = k$$ where \(a\), \(b\) and \(k\) are constants to be found. [3 marks] \item State the coordinates of \(P\) [1 mark] \item Find the \(y\)-coordinate of \(Q\) [2 marks] \item The line segment \(QR\) is a tangent to the circle as shown in Figure 2 below. **Figure 2** \includegraphics{figure_9d} The point \(R\) has coordinates \((9, -3)\). Find the angle \(QPR\) Give your answer in radians to three significant figures. [3 marks]
Edexcel AS Paper 1 Specimen Q17
10 marks Standard +0.3
A circle \(C\) with centre at \((-2, 6)\) passes through the point \((10, 11)\).
  1. Show that the circle \(C\) also passes through the point \((10, 1)\). [3]
The tangent to the circle \(C\) at the point \((10, 11)\) meets the \(y\) axis at the point \(P\) and the tangent to the circle \(C\) at the point \((10, 1)\) meets the \(y\) axis at the point \(Q\).
  1. Show that the distance \(PQ\) is \(58\) explaining your method clearly. [7]
Edexcel AS Paper 1 Q14
11 marks Standard +0.3
A curve with centre \(C\) has equation $$x^2 + y^2 + 2x - 6y - 40 = 0$$
    1. State the coordinates of \(C\).
    2. Find the radius of the circle, giving your answer as \(r = n\sqrt{2}\). [3]
  2. The line \(l\) is a tangent to the circle and has gradient \(-7\). Find two possible equations for \(l\), giving your answers in the form \(y = mx + c\). [8]
OCR PURE Q8
6 marks Standard +0.8
In this question you must show detailed reasoning. The lines \(y = \frac{1}{2}x\) and \(y = -\frac{1}{2}x\) are tangents to a circle at \((2, 1)\) and \((-2, 1)\) respectively. Find the equation of the circle in the form \(x^2 + y^2 + ax + by + c = 0\), where \(a\), \(b\) and \(c\) are constants. [6]
OCR PURE Q8
7 marks Challenging +1.2
In this question you must show detailed reasoning. A circle has equation \(x^2 + y^2 - 6x - 4y + 12 = 0\). Two tangents to this circle pass through the point \((0, 1)\). You are given that the scales on the \(x\)-axis and the \(y\)-axis are the same. Find the angle between these two tangents. [7]
OCR MEI AS Paper 2 2018 June Q8
7 marks Standard +0.3
In this question you must show detailed reasoning. The centre of a circle C is at the point \((-1, 3)\) and C passes through the point \((1, -1)\). The straight line L passes through the points \((1, 9)\) and \((4, 3)\). Show that L is a tangent to C. [7]
WJEC Unit 1 2019 June Q09
12 marks Moderate -0.3
The points \(A(-2, 4)\) and \(B(6, 10)\) are such that \(AB\) is the diameter of a circle.
  1. Show that the centre of the circle has coordinates \((2, 7)\). [1]
  2. The equation of the circle is \(x^2 + y^2 + ax + by + c = 0\). Determine the values of \(a\), \(b\), \(c\). [3]
A straight line, with equation \(y = x + 6\), passes through the point \(A\) and cuts the circle again at the point \(C\).
  1. Find the coordinates of \(C\). [5]
  2. Calculate the exact area of the triangle \(ABC\). [3]
WJEC Unit 1 2022 June Q7
11 marks Standard +0.3
A circle \(C\) has centre \(A\) and equation \(x^2 + y^2 - 4x - 6y = 3\).
  1. Find the coordinates of \(A\) and the radius of \(C\). [3]
The line \(L\) with equation \(y = x + 5\) intersects \(C\) at the points \(P\) and \(Q\).
  1. Determine the coordinates of \(P\) and \(Q\). [4]
  2. The point \(B\) is on \(PQ\) and is such that \(AB\) is perpendicular to \(PQ\). Find the length of \(PB\). [2]
  3. Show that the area of the smaller segment enclosed by \(C\) and \(L\) is \(4\pi - 8\). [2]
WJEC Unit 1 2023 June Q3
15 marks Moderate -0.3
The point \(A\) has coordinates \((-2, 5)\) and the point \(B\) has coordinates \((3, 8)\). The point \(C\) lies on the \(x\)-axis such that \(AC\) is perpendicular to \(AB\).
  1. Find the equation of \(AB\). [3]
  2. Show that \(C\) has coordinates \((1, 0)\). [3]
  3. Calculate the area of triangle \(ABC\). [4]
  4. Find the equation of the circle which passes through the points \(A\), \(B\) and \(C\). [5]
WJEC Unit 1 2024 June Q18
12 marks Standard +0.8
  1. A circle C has centre \((-3, -1)\) and radius \(\sqrt{5}\). Show that the equation of C can be written as \(x^2 + y^2 + 6x + 2y + 5 = 0\). [2]
    1. Find the equations of the tangents to C that pass through the origin O. [6]
    2. Determine the coordinates of the points where the tangents touch the circle. [4]
WJEC Unit 1 Specimen Q1
7 marks Moderate -0.8
The circle \(C\) has centre \(A\) and equation $$x^2 + y^2 - 2x + 6y - 15 = 0.$$
  1. Find the coordinates of \(A\) and the radius of \(C\). [3]
  2. The point \(P\) has coordinates \((4, -7)\) and lies on \(C\). Find the equation of the tangent to \(C\) at \(P\). [4]
SPS SPS FM 2019 Q7
7 marks Standard +0.3
\includegraphics{figure_7} The diagram shows a circle which passes through the points \(A(2, 9)\) and \(B(10, 3)\). \(AB\) is a diameter of the circle.
  1. The tangent to the circle at the point \(B\) cuts the \(x\)-axis at \(C\). Find the exact coordinates of \(C\). [4]
  2. Find the exact area of the triangle formed by \(B\), \(C\) and the centre of the circle [3]
SPS SPS FM 2020 October Q8
10 marks Moderate -0.8
The equation of a circle is \(x^2 + y^2 + 6x - 2y - 10 = 0\).
  1. Find the centre and radius of the circle. [3]
  2. Find the coordinates of any points where the line \(y = 2x - 3\) meets the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [4]
  3. State what can be deduced from the answer to part ii. about the line \(y = 2x - 3\) and the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [1]
  4. The point \(A(-1,5)\) lies on the circumference of the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). Given that \(AB\) is a diameter of the circle, find the coordinates of \(B\). [2]
SPS SPS SM 2020 October Q8
10 marks Moderate -0.8
The equation of a circle is \(x^2 + y^2 + 6x - 2y - 10 = 0\).
  1. Find the centre and radius of the circle. [3]
  2. Find the coordinates of any points where the line \(y = 2x - 3\) meets the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [4]
  3. State what can be deduced from the answer to part ii. about the line \(y = 2x - 3\) and the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). [1]
  4. The point \(A(-1,5)\) lies on the circumference of the circle \(x^2 + y^2 + 6x - 2y - 10 = 0\). Given that \(AB\) is a diameter of the circle, find the coordinates of \(B\). [2]
SPS SPS SM Pure 2021 June Q11
9 marks Standard +0.3
  1. A circle \(C_1\) has equation $$x^2 + y^2 + 18x - 2y + 30 = 0$$ The line \(l\) is the tangent to \(C_1\) at the point \(P(-5, 7)\). Find an equation of \(l\) in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers to be found. [5]
  2. A different circle \(C_2\) has equation $$x^2 + y^2 - 8x + 12y + k = 0$$ where \(k\) is a constant. Given that \(C_2\) lies entirely in the 4th quadrant, find the range of possible values for \(k\). [4]
SPS SPS SM Pure 2021 May Q6
7 marks Challenging +1.2
In this question you must show detailed reasoning. A circle touches the lines \(y = \frac{1}{2}x\) and \(y = 2x\) at \((6, 3)\) and \((3, 6)\) respectively. \includegraphics{figure_6} Find the equation of the circle. [7]
SPS SPS SM 2022 October Q9
7 marks Standard +0.3
In this question you must show detailed reasoning. The centre of a circle C is the point (-1, 3) and C passes through the point (1, -1). The straight line L passes through the points (1, 9) and (4, 3). Show that L is a tangent to C. [7]
SPS SPS SM Pure 2022 June Q3
10 marks Easy -1.2
A circle has centre \(C(3, -8)\) and radius \(10\).
  1. Express the equation of the circle in the form $$(x - a)^2 + (y - b)^2 = k$$ [2 marks]
  2. Find the \(x\)-coordinates of the points where the circle crosses the \(x\)-axis. [3 marks]
  3. The line with equation \(y = 2x + 1\) intersects the circle.
    1. Show that the \(x\)-coordinates of the points of intersection satisfy the equation $$x^2 + 6x - 2 = 0$$ [3 marks]
    2. Hence show that the \(x\)-coordinates of the points of intersection are of the form \(m \pm \sqrt{n}\), where \(m\) and \(n\) are integers. [2 marks]
SPS SPS SM 2022 October Q10
7 marks Standard +0.3
A circle \(C\) with radius \(r\)
  • lies only in the 1st quadrant
  • touches the \(x\)-axis and touches the \(y\)-axis
The line \(l\) has equation \(2x + y = 12\)
  1. Show that the \(x\) coordinates of the points of intersection of \(l\) with \(C\) satisfy $$5x^2 + (2r - 48)x + (r^2 - 24r + 144) = 0$$ [3]
Given also that \(l\) is a tangent to \(C\),
  1. find the two possible values of \(r\), giving your answers as fully simplified surds. [4]
SPS SPS SM Pure 2023 June Q8
7 marks Moderate -0.8
\includegraphics{figure_4} A circle with centre \((9, -6)\) touches the \(x\)-axis as shown in Figure 4.
  1. Write down an equation for the circle. [3] A line \(l\) is parallel to the \(x\)-axis. The line \(l\) cuts the circle at points \(P\) and \(Q\). Given that the distance \(PQ\) is 8
  2. find the two possible equations for \(l\). [4]
SPS SPS SM Pure 2023 September Q3
7 marks Moderate -0.8
$$x^2 + y^2 - 2x - 2y = 8$$ The circle with the above equation has radius \(r\) and has its centre at the point \(C\).
  1. Determine the value of \(r\) and the coordinates of \(C\). [3]
The point \(P(4,2)\) lies on the circle.
  1. Show that an equation of the tangent to the circle at \(P\) is [4] $$y = 14 - 3x.$$
SPS SPS SM 2023 October Q10
7 marks Standard +0.8
In this question you must show detailed reasoning. A circle has equation \(x^2 + y^2 - 6x - 4y + 12 = 0\). Two tangents to this circle pass through the point \((0, 1)\). You are given that the scales on the \(x\)-axis and the \(y\)-axis are the same. Find the angle between these two tangents. [7]
SPS SPS FM 2024 October Q9
9 marks Standard +0.8
A circle has centre \(C\) which lies on the \(x\)-axis, as shown in the diagram. The line \(y = x\) meets the circle at \(A\) and \(B\). The midpoint of \(AB\) is \(M\). \includegraphics{figure_9} The equation of the circle is \(x^2 - 6x + y^2 + a = 0\), where \(a\) is a constant.
  1. In this question you must show detailed reasoning. Find the \(x\)-coordinate of \(M\) and hence show that the area of triangle \(ABC\) is \(\frac{3}{2}\sqrt{9 - 2a}\). [6]
    1. Find the value of \(a\) when the area of triangle \(ABC\) is zero. [1]
    2. Give a geometrical interpretation of the case in part (b)(i). [1]
  3. Give a geometrical interpretation of the case where \(a = 5\). [1]
SPS SPS SM 2024 October Q8
5 marks Standard +0.3
A circle, C, has equation \(x^2 - 6x + y^2 = 16\). A second circle, D, has the following properties:
  • The line through the centres of circle C and circle D has gradient 1.
  • Circle D touches circle C at exactly one point.
  • The centre of circle D lies in the first quadrant.
  • Circle D has the same radius as circle C.
Find the coordinates of the centre of circle D. [5]
SPS SPS SM 2024 October Q5
11 marks Moderate -0.3
A line has equation \(y = 2x\) and a circle has equation \(x^2 + y^2 + 2x - 16y + 56 = 0\).
  1. Show that the line does not meet the circle. [3]
    1. Find the equation of the line through the centre of the circle that is perpendicular to the line \(y = 2x\). [4]
    2. Hence find the shortest distance between the line \(y = 2x\) and the circle, giving your answer in an exact form. [4]