Circle from diameter endpoints

Given two points that form a diameter, find the circle equation using the midpoint as centre and half the distance as radius.

45 questions · Moderate -0.5

1.03d Circles: equation (x-a)^2+(y-b)^2=r^2
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Edexcel Paper 2 Specimen Q9
9 marks Standard +0.3
  1. A circle with centre \(A ( 3 , - 1 )\) passes through the point \(P ( - 9,8 )\) and the point \(Q ( 15 , - 10 )\)
    1. Show that \(P Q\) is a diameter of the circle.
    2. Find an equation for the circle.
    A point \(R\) also lies on the circle. Given that the length of the chord \(P R\) is 20 units,
  2. find the length of the shortest distance from \(A\) to the chord \(P R\). Give your answer as a surd in its simplest form.
  3. Find the size of angle \(A R Q\), giving your answer to the nearest 0.1 of a degree.
AQA C1 2016 June Q5
13 marks Moderate -0.3
5 A circle with centre \(C ( 5 , - 3 )\) passes through the point \(A ( - 2,1 )\).
  1. Find the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
  2. Given that \(A B\) is a diameter of the circle, find the coordinates of the point \(B\).
  3. Find an equation of the tangent to the circle at the point \(A\), giving your answer in the form \(p x + q y + r = 0\), where \(p , q\) and \(r\) are integers.
  4. The point \(T\) lies on the tangent to the circle at \(A\) such that \(A T = 4\). Find the length of \(C T\).
Edexcel C1 Q7
13 marks Moderate -0.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8bae58f7-c53a-43ed-9a1d-2f718bd1e539-3_563_570_785_561} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The points \(A ( - 3 , - 2 )\) and \(B ( 8,4 )\) are at the ends of a diameter of the circle shown in Fig. 1.
  1. Find the coordinates of the centre of the circle.
  2. Find an equation of the diameter \(A B\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
  3. Find an equation of tangent to the circle at \(B\). The line \(l\) passes through \(A\) and the origin.
  4. Find the coordinates of the point at which \(l\) intersects the tangent to the circle at \(B\), giving your answer as exact fractions.
Edexcel C2 Q7
9 marks Moderate -0.8
7. The points \(P\) and \(Q\) have coordinates \(( - 2,6 )\) and \(( 4 , - 1 )\) respectively. Given that \(P Q\) is a diameter of circle \(C\),
  1. find the coordinates of the centre of \(C\),
  2. show that \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 2 x - 5 y - 14 = 0 .$$ The point \(R\) has coordinates (2, 7).
  3. Show that \(R\) lies on \(C\) and hence, state the size of \(\angle P R Q\) in degrees.
Edexcel C2 Q8
11 marks Moderate -0.3
  1. The point \(A\) has coordinates ( 4,6 ).
Given that \(O A\), where \(O\) is the origin, is a diameter of circle \(C\),
  1. find an equation for \(C\). Circle \(C\) crosses the \(x\)-axis at \(O\) and at the point \(B\).
  2. Find the coordinates of \(B\).
  3. Find an equation for the tangent to \(C\) at \(B\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
OCR MEI C1 2009 January Q11
14 marks Moderate -0.3
  1. Show that the equation of the circle with AB as diameter may be written as $$( x - 5 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 40$$
  2. Find the coordinates of the points of intersection of this circle with the \(y\)-axis. Give your answer in the form \(a \pm \sqrt { b }\).
  3. Find the equation of the tangent to the circle at B . Hence find the coordinates of the points of intersection of this tangent with the axes.
OCR AS Pure 2017 Specimen Q2
5 marks Moderate -0.8
2 Points \(A\) and \(B\) have coordinates \(( 3,0 )\) and \(( 9,8 )\) respectively. The line \(A B\) is a diameter of a circle.
  1. Find the coordinates of the centre of the circle.
  2. Find the equation of the tangent to the circle at the point \(B\).
Edexcel M2 Q16
13 marks Moderate -0.8
16. \section*{Figure 3}
\includegraphics[max width=\textwidth, alt={}]{90893903-4f36-4974-8eaa-0f462f35f442-08_581_575_395_609}
The points \(A ( - 3 , - 2 )\) and \(B ( 8,4 )\) are at the ends of a diameter of the circle shown in Fig. 3.
  1. Find the coordinates of the centre of the circle.
  2. Find an equation of the diameter \(A B\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
  3. Find an equation of tangent to the circle at \(B\). The line \(l\) passes through \(A\) and the origin.
  4. Find the coordinates of the point at which \(l\) intersects the tangent to the circle at \(B\), giving your answer as exact fractions.
AQA AS Paper 2 2019 June Q7
6 marks Moderate -0.3
7 The points \(A ( a , 3 )\) and \(B ( 10,6 )\) lie on a circle. \(A B\) is a diameter of the circle and passes through the point ( 2,4 )
The circle has equation $$( x - c ) ^ { 2 } + ( y - d ) ^ { 2 } = e$$ where \(c , d\) and \(e\) are rational numbers. Find the values of \(a , c , d\) and \(e\).
Edexcel C1 Q16
13 marks Standard +0.3
\includegraphics{figure_3} The points \(A(-3, -2)\) and \(B(8, 4)\) are at the ends of a diameter of the circle shown in Fig. 3.
  1. Find the coordinates of the centre of the circle. [2]
  2. Find an equation of the diameter \(AB\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
  3. Find an equation of tangent to the circle at \(B\). [3]
The line \(l\) passes through \(A\) and the origin.
  1. Find the coordinates of the point at which \(l\) intersects the tangent to the circle at \(B\), giving your answer as exact fractions. [4]
Edexcel C2 Q2
6 marks Moderate -0.8
The points \(A\) and \(B\) have coordinates \((5, -1)\) and \((13, 11)\) respectively.
  1. Find the coordinates of the mid-point of \(AB\). [2]
Given that \(AB\) is a diameter of the circle \(C\),
  1. find an equation for \(C\). [4]
Edexcel C2 Q3
7 marks Moderate -0.8
\includegraphics{figure_1} In Figure 1, \(A(4, 0)\) and \(B(3, 5)\) are the end points of a diameter of the circle \(C\). Find
  1. the exact length of \(AB\), [2]
  2. the coordinates of the midpoint \(P\) of \(AB\), [2]
  3. an equation for the circle \(C\). [3]
Edexcel C1 Q7
13 marks Moderate -0.3
\includegraphics{figure_1} The points \(A(-3, -2)\) and \(B(8, 4)\) are at the ends of a diameter of the circle shown in Fig. 1.
  1. Find the coordinates of the centre of the circle. [2]
  2. Find an equation of the diameter \(AB\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
  3. Find an equation of tangent to the circle at \(B\). [3]
The line \(l\) passes through \(A\) and the origin.
  1. Find the coordinates of the point at which \(l\) intersects the tangent to the circle at \(B\), giving your answer as exact fractions. [4]
OCR C1 2006 June Q9
12 marks Easy -1.2
The points \(A\) and \(B\) have coordinates \((4, -2)\) and \((10, 6)\) respectively. \(C\) is the mid-point of \(AB\). Find
  1. the coordinates of \(C\), [2]
  2. the length of \(AC\), [2]
  3. the equation of the circle that has \(AB\) as a diameter, [3]
  4. the equation of the tangent to the circle in part (iii) at the point \(A\), giving your answer in the form \(ax + by = c\). [5]
Edexcel C2 Q2
4 marks Moderate -0.5
The point \(A\) has coordinates \((2, 5)\) and the point \(B\) has coordinates \((-2, 8)\). Find, in cartesian form, an equation of the circle with diameter \(AB\). [4]
Edexcel C2 Q1
4 marks Moderate -0.8
The point \(A\) has coordinates \((2, 5)\) and the point \(B\) has coordinates \((-2, 8)\). Find, in cartesian form, an equation of the circle with diameter \(AB\). [4]
OCR H240/02 2020 November Q7
8 marks Moderate -0.8
\(A\) and \(B\) are fixed points in the \(x\)-\(y\) plane. The position vectors of \(A\) and \(B\) are \(\mathbf{a}\) and \(\mathbf{b}\) respectively. State, with reference to points \(A\) and \(B\), the geometrical significance of
  1. the quantity \(|\mathbf{a} - \mathbf{b}|\), [1]
  2. the vector \(\frac{1}{2}(\mathbf{a} + \mathbf{b})\). [1]
The circle \(P\) is the set of points with position vector \(\mathbf{p}\) in the \(x\)-\(y\) plane which satisfy $$\left|\mathbf{p} - \frac{1}{2}(\mathbf{a} + \mathbf{b})\right| = \frac{1}{2}|\mathbf{a} - \mathbf{b}|.$$
  1. State, in terms of \(\mathbf{a}\) and \(\mathbf{b}\),
    1. the position vector of the centre of \(P\), [1]
    2. the radius of \(P\). [1]
It is now given that \(\mathbf{a} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}\), \(\mathbf{b} = \begin{pmatrix} 4 \\ 5 \end{pmatrix}\) and \(\mathbf{p} = \begin{pmatrix} x \\ y \end{pmatrix}\).
  1. Find a cartesian equation of \(P\). [4]
OCR H240/03 2022 June Q3
4 marks Moderate -0.8
The points \(P\) and \(Q\) have coordinates \((2, -5)\) and \((3, 1)\) respectively. Determine the equation of the circle that has \(PQ\) as a diameter. Give your answer in the form \(x^2 + y^2 + ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
AQA AS Paper 1 2022 June Q6
9 marks Moderate -0.3
\(AB\) is a diameter of a circle where \(A\) is \((1, 4)\) and \(B\) is \((7, -2)\)
  1. Find the coordinates of the midpoint of \(AB\). [1 mark]
  2. Show that the equation of the circle may be written as $$x^2 + y^2 - 8x - 2y = 1$$ [4 marks]
  3. The circle has centre \(C\) and crosses the \(x\)-axis at points \(D\) and \(E\). Find the exact area of triangle \(DEC\). [4 marks]
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]