Find centre and radius from equation

Given a circle equation in expanded form, complete the square to find the centre coordinates and radius.

21 questions · Moderate -1.0

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Edexcel C12 2015 January Q7
8 marks Moderate -0.8
  1. The circle \(C\) has equation
$$x ^ { 2 } + y ^ { 2 } + 10 x - 6 y + 18 = 0$$ Find
  1. the coordinates of the centre of \(C\),
  2. the radius of \(C\). The circle \(C\) meets the line with equation \(x = - 3\) at two points.
  3. Find the exact values for the \(y\) coordinates of these two points, giving your answers as fully simplified surds.
Edexcel C2 2011 June Q4
8 marks Moderate -0.8
4. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 4 x - 2 y - 11 = 0$$ Find
  1. the coordinates of the centre of \(C\),
  2. the radius of \(C\),
  3. the coordinates of the points where \(C\) crosses the \(y\)-axis, giving your answers as simplified surds.
Edexcel C2 2017 June Q5
7 marks Moderate -0.8
5. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 10 x + 6 y + 30 = 0$$ Find
  1. the coordinates of the centre of \(C\),
  2. the radius of \(C\),
  3. the \(y\) coordinates of the points where the circle \(C\) crosses the line with equation \(x = 4\), giving your answers as simplified surds.
Edexcel C12 2017 January Q2
7 marks Moderate -0.8
A circle, with centre \(C\) and radius \(r\), has equation $$x ^ { 2 } + y ^ { 2 } - 8 x + 4 y - 12 = 0$$ Find
  1. the coordinates of \(C\),
  2. the exact value of \(r\). The circle cuts the \(y\)-axis at the points \(A\) and \(B\).
  3. Find the coordinates of the points \(A\) and \(B\).
OCR MEI C1 Q6
4 marks Easy -1.2
6 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } - 2 x - 8 = 0\).
Find the centre and radius of the circle.
OCR C1 Q3
4 marks Moderate -0.8
3. A circle has the equation $$x ^ { 2 } + y ^ { 2 } - 6 y - 7 = 0$$
  1. Find the coordinates of the centre of the circle.
  2. Find the radius of the circle.
OCR C1 2011 January Q9
15 marks Standard +0.3
9 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 2 y - 3 = 0\).
  1. Find the coordinates of \(C\) and the radius of the circle.
  2. Find the values of \(k\) for which the line \(y = k\) is a tangent to the circle, giving your answers in simplified surd form.
  3. The points \(S\) and \(T\) lie on the circumference of the circle. \(M\) is the mid-point of the chord \(S T\). Given that the length of \(C M\) is 2 , calculate the length of the chord \(S T\).
  4. Find the coordinates of the point where the circle meets the line \(x - 2 y - 12 = 0\).
OCR H240/03 2018 June Q1
3 marks Easy -1.2
1 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 8 x - 2 y - 7 = 0\).
Find
  1. the coordinates of \(C\),
  2. the radius of the circle.
OCR PURE 2066 Q2
4 marks Moderate -0.5
2 The circle \(x ^ { 2 } + y ^ { 2 } - 4 x + k y + 12 = 0\) has radius 1.
Find the two possible values of the constant \(k\).
OCR PURE 2022 June Q4
3 marks Moderate -0.5
4 The circle \(x ^ { 2 } + y ^ { 2 } - 6 x + 4 y + k = 0\) has radius 5.
Determine the value of \(k\).
OCR MEI AS Paper 2 2019 June Q4
3 marks Easy -1.2
4 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } + 8 x - 6 y - 39 = 0\).
  1. Find the coordinates of the centre of the circle.
  2. Find the radius of the circle.
OCR MEI Paper 2 2023 June Q2
3 marks Easy -1.2
2 The equation of a circle is
\(x ^ { 2 } - 12 x + y ^ { 2 } + 8 y + 3 = 0\).
  1. Find the radius of the circle.
  2. State the coordinates of the centre of the circle.
AQA C1 2013 June Q3
12 marks Moderate -0.8
3 A circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 10 x + 14 y + 25 = 0$$
  1. Write the equation of \(C\) in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$ where \(a , b\) and \(k\) are integers.
  2. Hence, for the circle \(C\), write down:
    1. the coordinates of its centre;
    2. its radius.
    1. Sketch the circle \(C\).
    2. Write down the coordinates of the point on \(C\) that is furthest away from the \(x\)-axis.
  4. Given that \(k\) has the same value as in part (a), describe geometrically the transformation which maps the circle with equation \(( x + 1 ) ^ { 2 } + y ^ { 2 } = k\) onto the circle \(C\).
Edexcel C2 Q1
4 marks Easy -1.2
  1. A circle \(C\) has equation
$$x ^ { 2 } + y ^ { 2 } - 10 x + 6 y - 15 = 0 .$$
  1. Find the coordinates of the centre of \(C\).
  2. Find the radius of \(C\).
Edexcel C2 Q1
4 marks Easy -1.2
  1. A circle \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 10 x + 6 y - 15 = 0\).
    1. Find the coordinates of the centre of \(C\).
    2. Find the radius of \(C\).
      [0pt] [P3 June 2001 Question 1]
    3. \(\mathrm { f } ( x ) \equiv a x ^ { 3 } + b x ^ { 2 } - 7 x + 14\), where \(a\) and \(b\) are constants.
    Given that when \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\) the remainder is 9 ,
  2. write down an equation connecting \(a\) and \(b\). Given also that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\),
  3. find the values of \(a\) and \(b\).
    [0pt] [P3 June 2001 Question 2]
Edexcel C2 Q1
4 marks Easy -1.2
  1. A circle has the equation \(x ^ { 2 } + y ^ { 2 } - 6 y - 7 = 0\).
    1. Find the coordinates of the centre of the circle.
    2. Find the radius of the circle.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{da9672c1-d1d8-4c2f-89ef-2055da37f720-2_369_689_552_511} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows the sector \(O A B\) of a circle, centre \(O\), in which \(\angle A O B = 2.5\) radians.
    Given that the perimeter of the sector is 36 cm ,
  2. find the length \(O A\),
  3. find the area of the shaded segment.
Edexcel C2 Q6
10 marks Moderate -0.8
6. The circle \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 12 x + 8 y + 16 = 0 .$$
  1. Find the coordinates of the centre of \(C\).
  2. Find the radius of \(C\).
  3. Sketch C. Given that \(C\) crosses the \(x\)-axis at the points \(A\) and \(B\),
  4. find the length \(A B\), giving your answer in the form \(k \sqrt { 5 }\).
OCR Pure 1 2018 March Q1
4 marks Easy -1.2
1 A circle with equation \(x ^ { 2 } + y ^ { 2 } + 6 x - 4 y = k\) has a radius of 4 .
  1. Find the coordinates of the centre of the circle.
  2. Find the value of the constant \(k\).
Edexcel C2 Q11
4 marks Easy -1.2
11. A circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 10 x + 6 y - 15 = 0$$
  1. Find the coordinates of the centre of \(C\).
  2. Find the radius of \(C\).
    [0pt] [P3 June 2001 Question 1]
AQA AS Paper 2 2019 June Q2
1 marks Easy -1.3
2 Find the centre of the circle \(x ^ { 2 } + y ^ { 2 } + 4 x - 6 y = 12\)
Tick ( \(\checkmark\) ) one box.
(-2, -3) □
(-2, 3) □
\(( 2 , - 3 )\) □
\(( 2,3 )\) □
AQA Paper 2 2023 June Q3
1 marks Easy -2.0
3 A circle has equation $$( x - 5 ) ^ { 2 } + ( y - 13 ) ^ { 2 } = 16$$ Find the radius of the circle. Circle your answer. 41216256