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 C2 Q1
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]
Edexcel C2 Q1
4 marks Moderate -0.8
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]
Edexcel C2 Q1
4 marks Moderate -0.8
A circle has the equation \(x^2 + y^2 - 6y - 7 = 0\).
  1. Find the coordinates of the centre of the circle. [2]
  2. Find the radius of the circle. [2]
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]
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]