Circle through three points using right angle in semicircle

Find the equation of a circle through three points by identifying a right angle (angle in semicircle) to determine the diameter, then finding centre and radius.

12 questions · Moderate -0.4

1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle
Sort by: Default | Easiest first | Hardest first
CAIE P1 2021 June Q10
10 marks Moderate -0.3
10 Points \(A ( - 2,3 ) , B ( 3,0 )\) and \(C ( 6,5 )\) lie on the circumference of a circle with centre \(D\).
  1. Show that angle \(A B C = 90 ^ { \circ }\).
  2. Hence state the coordinates of \(D\).
  3. Find an equation of the circle.
    The point \(E\) lies on the circumference of the circle such that \(B E\) is a diameter.
  4. Find an equation of the tangent to the circle at \(E\).
CAIE P1 2021 March Q8
7 marks Moderate -0.8
8 The points \(A ( 7,1 ) , B ( 7,9 )\) and \(C ( 1,9 )\) are on the circumference of a circle.
  1. Find an equation of the circle.
  2. Find an equation of the tangent to the circle at \(B\).
CAIE P1 2022 November Q11
11 marks Moderate -0.3
11 The coordinates of points \(A , B\) and \(C\) are \(A ( 5 , - 2 ) , B ( 10,3 )\) and \(C ( 2 p , p )\), where \(p\) is a constant.
  1. Given that \(A C\) and \(B C\) are equal in length, find the value of the fraction \(p\).
  2. It is now given instead that \(A C\) is perpendicular to \(B C\) and that \(p\) is an integer.
    1. Find the value of \(p\).
    2. Find the equation of the circle which passes through \(A , B\) and \(C\), giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\), where \(a , b\) and \(c\) are constants.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
Edexcel P2 2022 January Q6
8 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{59c9f675-e7eb-47b9-b233-dfbe1844f792-18_579_620_219_667} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The points \(P ( 23,14 ) , Q ( 15 , - 30 )\) and \(R ( - 7 , - 26 )\) lie on the circle \(C\), as shown in Figure 1.
  1. Show that angle \(P Q R = 90 ^ { \circ }\)
  2. Hence, or otherwise, find
    1. the centre of \(C\),
    2. the radius of \(C\). Given that the point \(S\) lies on \(C\) such that the distance \(Q S\) is greatest,
  3. find an equation of the tangent to \(C\) at \(S\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found.
OCR C1 2005 January Q10
13 marks Moderate -0.8
10 The points \(D , E\) and \(F\) have coordinates \(( - 2,0 ) , ( 0 , - 1 )\) and \(( 2,3 )\) respectively.
  1. Calculate the gradient of \(D E\).
  2. Find the equation of the line through \(F\), parallel to \(D E\), giving your answer in the form \(a x + b y + c = 0\).
  3. By calculating the gradient of \(E F\), show that \(D E F\) is a right-angled triangle.
  4. Calculate the length of \(D F\).
  5. Use the results of parts (iii) and (iv) to show that the circle which passes through \(D , E\) and \(F\) has equation \(x ^ { 2 } + y ^ { 2 } - 3 y - 4 = 0\).
OCR MEI AS Paper 2 2021 November Q10
6 marks Standard +0.3
  1. Show that PQ is perpendicular to QR . A circle passes through \(\mathrm { P } , \mathrm { Q }\) and R .
  2. Determine the coordinates of the centre of the circle.
CAIE P1 2023 November Q11
10 marks Standard +0.3
The coordinates of points \(A\), \(B\) and \(C\) are \((6, 4)\), \((p, 7)\) and \((14, 18)\) respectively, where \(p\) is a constant. The line \(AB\) is perpendicular to the line \(BC\).
  1. Given that \(p < 10\), find the value of \(p\). [4]
A circle passes through the points \(A\), \(B\) and \(C\).
  1. Find the equation of the circle. [3]
  2. Find the equation of the tangent to the circle at \(C\), giving the answer in the form \(dx + ey + f = 0\), where \(d\), \(e\) and \(f\) are integers. [3]
OCR MEI C1 2006 June Q11
12 marks Moderate -0.8
A\((9, 8)\), B\((5, 0)\) and C\((3, 1)\) are three points.
  1. Show that AB and BC are perpendicular. [3]
  2. Find the equation of the circle with AC as diameter. You need not simplify your answer. Show that B lies on this circle. [6]
  3. BD is a diameter of the circle. Find the coordinates of D. [3]
OCR C1 Q6
9 marks Moderate -0.8
The points \(P\) and \(Q\) have coordinates \((-2, 6)\) and \((4, -1)\) respectively. Given that \(PQ\) is a diameter of circle \(C\),
  1. find the coordinates of the centre of \(C\), [2]
  2. show that \(C\) has the equation $$x^2 + y^2 - 2x - 5y - 14 = 0. \quad [5]$$
The point \(R\) has coordinates \((2, 7)\).
  1. Show that \(R\) lies on \(C\) and hence, state the size of \(\angle PRQ\) in degrees. [2]
OCR C1 Q6
10 marks Moderate -0.3
The points \(P\), \(Q\) and \(R\) have coordinates \((-5, 2)\), \((-3, 8)\) and \((9, 4)\) respectively.
  1. Show that \(\angle PQR = 90°\). [4]
Given that \(P\), \(Q\) and \(R\) all lie on a circle,
  1. find the coordinates of the centre of the circle, [3]
  2. show that the equation of the circle can be written in the form $$x^2 + y^2 - 4x - 6y = k,$$ where \(k\) is an integer to be found. [3]
OCR MEI C1 Q2
12 marks Moderate -0.8
A\((9, 8)\), B\((5, 0)\) and C\((3, 1)\) are three points.
  1. Show that AB and BC are perpendicular. [3]
  2. Find the equation of the circle with AC as diameter. You need not simplify your answer. Show that B lies on this circle. [6]
  3. BD is a diameter of the circle. Find the coordinates of D. [3]
OCR MEI C1 Q6
13 marks Moderate -0.3
The points A \((-1, 6)\), B \((1, 0)\) and C \((13, 4)\) are joined by straight lines.
  1. Prove that the lines AB and BC are perpendicular. [3]
  2. Find the area of triangle ABC. [3]
  3. A circle passes through the points A, B and C. Justify the statement that AC is a diameter of this circle. Find the equation of this circle. [6]
  4. Find the coordinates of the point on this circle that is furthest from B. [1]