Find centre and radius from equation

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

31 questions · Easy -1.0

1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle
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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 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 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 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 H240/01 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 \(k\).
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
Pre-U Pre-U 9794/1 2012 Specimen Q1
2 marks Easy -1.8
1 Write down the coordinates of the centre and the radius of the circle with equation $$( x + 5 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 36 .$$
Pre-U Pre-U 9794/1 2015 June Q5
9 marks Moderate -0.8
5 A circle has equation \(x ^ { 2 } - 6 x + y ^ { 2 } - 4 y = 12\).
  1. Show that the centre of the circle is at the point \(( 3,2 )\) and find the radius.
  2. \(P Q\) is a diameter of the circle where \(P\) has coordinates \(( - 1 , - 1 )\). Find the equation of \(P Q\), giving your answer in the form \(a x + b y = c\) where \(a , b\) and \(c\) are integers.
  3. Another diameter of the circle passes through the point \(( 0,6 )\). Show that this diameter is perpendicular to \(P Q\).
Pre-U Pre-U 9794/1 2016 June Q5
4 marks Moderate -0.8
5 The circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x - k = 0\) has radius 5 . Find the coordinates of the centre of the circle and the value of \(k\).
Pre-U Pre-U 9794/1 2016 Specimen Q1
4 marks Easy -1.8
1 A circle has equation \(( x - 4 ) ^ { 2 } + ( y + 7 ) ^ { 2 } = 64\).
  1. Write down the coordinates of the centre and the radius of the circle. Two points, \(A\) and \(B\), lie on the circle and have coordinates \(( 4,1 )\) and \(( 12 , - 7 )\) respectively.
  2. Find the coordinates of the midpoint of the chord \(A B\).
Pre-U Pre-U 9794/1 2017 June Q1
3 marks Easy -1.8
1 The equation of a circle is given by \(( x - 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = r ^ { 2 }\).
  1. Write down the coordinates of the centre of the circle.
  2. The circle passes through the point \(( 0,2 )\). Find the length of the diameter.
Pre-U Pre-U 9794/1 2019 Specimen Q1
2 marks Easy -1.8
1 A circle has equation \(( x - 4 ) ^ { 2 } + ( y + 7 ) ^ { 2 } = 64\).
  1. Write down the coordinates of the centre and the radius of the circle. Two points, \(A\) and \(B\), lie on the circle and have coordinates \(( 4,1 )\) and \(( 12 , - 7 )\) respectively.
  2. Find the coordinates of the midpoint of the chord \(A B\).
Edexcel C2 Q11
4 marks Easy -1.2
A circle \(C\) has equation $$x^2 + y^2 - 10x + 6y - 15 = 0.$$
  1. Find the coordinates of the centre of \(C\). [2]
  2. Find the radius of \(C\). [2]
OCR C1 2013 June Q6
5 marks Moderate -0.8
A circle \(C\) has equation \(x^2 + y^2 + 8y - 24 = 0\).
  1. Find the centre and radius of the circle. [3]
  2. The point \(A(2, 2)\) lies on the circumference of \(C\). Given that \(AB\) is a diameter of the circle, find the coordinates of \(B\). [2]