1.03d Circles: equation (x-a)^2+(y-b)^2=r^2

424 questions

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CAIE P1 2021 November Q6
7 marks Standard +0.3
  1. Find the perimeter of the plate, giving your answer in terms of \(\pi\).
  2. Find the area of the plate, giving your answer in terms of \(\pi\) and \(\sqrt { 3 }\).
CAIE P1 2022 November Q10
8 marks Standard +0.3
  1. Find the perimeter of the cross-section RASB, giving your answer correct to 2 decimal places.
  2. Find the difference in area of the two triangles \(A O B\) and \(A P B\), giving your answer correct to 2 decimal places.
  3. Find the area of the cross-section RASB, giving your answer correct to 1 decimal place.
CAIE P1 2022 November Q10
10 marks Standard +0.3
  1. Find the coordinates of \(A\).
  2. Find the volume of revolution when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Give your answer in the form \(\frac { \pi } { a } ( b \sqrt { c } - d )\), where \(a , b , c\) and \(d\) are integers.
  3. Find an exact expression for the perimeter of the shaded region.
CAIE P1 2017 June Q4
7 marks Moderate -0.8
  1. Express the perimeter of the shaded region in terms of \(r\) and \(\theta\).
  2. In the case where \(r = 5\) and \(\theta = \frac { 1 } { 6 } \pi\), find the area of the shaded region.
CAIE P1 2017 March Q4
6 marks Standard +0.3
  1. Show that angle \(C B D = \frac { 9 } { 14 } \pi\) radians.
  2. Find the perimeter of the shaded region.
CAIE P1 2016 November Q4
6 marks Easy -1.2
  1. Find the equation of the line \(C D\), giving your answer in the form \(y = m x + c\).
  2. Find the distance \(A D\).
AQA C1 2014 June Q7
14 marks Moderate -0.5
  1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
    1. Write down the coordinates of \(C\).
    2. Show that the circle has radius \(n \sqrt { 5 }\), where \(n\) is an integer.
  3. Find the equation of the tangent to the circle at the point \(A\), giving your answer in the form \(x + p y = q\), where \(p\) and \(q\) are integers.
  4. The point \(B\) lies on the tangent to the circle at \(A\) and the length of \(B C\) is 6. Find the length of \(A B\).
    [0pt] [3 marks]
    \includegraphics[max width=\textwidth, alt={}]{f2124c89-79de-4758-b7b8-ff273345b9dd-8_1421_1709_1286_153}
AQA C1 2015 June Q4
11 marks Standard +0.3
  1. Express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = d$$
    1. State the coordinates of \(C\).
    2. Find the radius of the circle, giving your answer in the form \(n \sqrt { 2 }\).
  3. The point \(P\) with coordinates \(( 4 , k )\) lies on the circle. Find the possible values of \(k\).
  4. The points \(Q\) and \(R\) also lie on the circle, and the length of the chord \(Q R\) is 2 . Calculate the shortest distance from \(C\) to the chord \(Q R\).
    [0pt] [2 marks]
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 MEI C1 Q10
12 marks Moderate -0.3
  1. Write down the equations of the circles A and B .
  2. Find the \(x\) coordinates of the points where the two curves intersect.
  3. Find the \(y\) coordinates of these points, giving your answers in surd form.
OCR H240/01 2020 November Q11
10 marks Challenging +1.2
    1. Show that the \(x\)-coordinate of \(A\) satisfies the equation \(\left( m ^ { 2 } + 1 \right) x ^ { 2 } - 10 ( m + 1 ) x + 40 = 0\).
    2. Hence determine the equation of the tangent to the circle at \(A\) which passes through \(P\). [4] A second tangent is drawn from \(P\) to meet the circle at a second point \(B\). The equation of this tangent is of the form \(y = n x + 2\), where \(n\) is a constant less than 1 .
  2. Determine the exact value of \(\tan A P B\).
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.
OCR H240/02 2021 November Q5
8 marks Standard +0.3
5 In this question you must show detailed reasoning. Points \(A , B\) and \(C\) have coordinates \(( 0,6 ) , ( 7,5 )\) and \(( 6 , - 2 )\) respectively.
  1. Find an equation of the perpendicular bisector of \(A B\).
  2. Hence, or otherwise, find an equation of the circle that passes through points \(A , B\) and \(C\).
OCR PURE Q6
6 marks Moderate -0.3
6 In this question you must show detailed reasoning.
\includegraphics[max width=\textwidth, alt={}]{7fc02f90-8f8b-4153-bba1-dc0807124e96-4_650_661_1765_242}
The diagram shows the line \(3 y + x = 7\) which is a tangent to a circle with centre \(( 3 , - 2 )\).
Find an equation for the circle.
OCR MEI Paper 3 Specimen Q5
5 marks Standard +0.3
5 In this question you must show detailed reasoning. Fig. 5 shows the circle with equation \(( x - 4 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 10\).
The points \(( 1,0 )\) and \(( 7,0 )\) lie on the circle. The point C is the centre of the circle. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b4e10fd2-4144-4019-bf00-070f93a2b05d-05_878_1000_685_255} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure} Find the area of the part of the circle below the \(x\)-axis.
OCR MEI C1 2007 January Q11
12 marks Moderate -0.3
11 There is an insert for use in this question. The graph of \(y = x + \frac { 1 } { x }\) is shown on the insert. The lowest point on one branch is \(( 1,2 )\). The highest point on the other branch is \(( - 1 , - 2 )\).
  1. Use the graph to solve the following equations, showing your method clearly. $$\text { (A) } x + \frac { 1 } { x } = 4$$ $$\text { (B) } 2 x + \frac { 1 } { x } = 4$$
  2. The equation \(( x - 1 ) ^ { 2 } + y ^ { 2 } = 4\) represents a circle. Find in exact form the coordinates of the points of intersection of this circle with the \(y\)-axis.
  3. State the radius and the coordinates of the centre of this circle. Explain how these can be used to deduce from the graph that this circle touches one branch of the curve \(y = x + \frac { 1 } { x }\) but does not intersect with the other.
OCR MEI C1 Q4
12 marks Moderate -0.8
4 There is an insert for use in this question. The graph of \(y = x + \frac { 1 } { x }\) is shown on the insert. The lowest point on one branch is \(( 1,2 )\). The highest point on the other branch is \(( - 1 , - 2 )\).
  1. Use the graph to solve the following equations, showing your method clearly.
    (A) \(x + \frac { 1 } { x } = 4\) (B) \(2 x + \frac { 1 } { x } = 4\)
  2. The equation \(( x - 1 ) ^ { 2 } + y ^ { 2 } = 4\) represents a circle. Find in exact form the coordinates of the points of intersection of this circle with the \(y\)-axis.
  3. State the radius and the coordinates of the centre of this circle. Explain how these can be used to deduce from the graph that this circle touches one branch of the curve \(y = x + \frac { 1 } { x }\) but does not intersect with the other.
OCR MEI AS Paper 2 2020 November Q7
8 marks Moderate -0.3
7 In this question you must show detailed reasoning. A circle has centre \(( 2 , - 1 )\) and radius 5. A straight line passes through the points \(( 1,1 )\) and \(( 9,5 )\).
Find the coordinates of the points of intersection of the line and the circle.
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 \(k\).
OCR H240/01 2018 September Q7
11 marks Moderate -0.3
7 A line has equation \(y = 2 x\) and a circle has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 16 y + 56 = 0\).
  1. Show that the line does not meet the circle.
  2. (a) Find the equation of the line through the centre of the circle that is perpendicular to the line \(y = 2 x\).
    (b) Hence find the shortest distance between the line \(y = 2 x\) and the circle, giving your answer in an exact form.
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 C1 Q5
Moderate -0.5
5. Solve the simultaneous equations $$\begin{gathered} x - 2 y = 1 \\ x ^ { 2 } + y ^ { 2 } = 29 \end{gathered}$$
Edexcel M2 Q3
11 marks Moderate -0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{90893903-4f36-4974-8eaa-0f462f35f442-02_650_1043_367_317} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The points \(A ( 3,0 )\) and \(B ( 0,4 )\) are two vertices of the rectangle \(A B C D\), as shown in Fig. 2.
  1. Write down the gradient of \(A B\) and hence the gradient of \(B C\). The point \(C\) has coordinates \(( 8 , k )\), where \(k\) is a positive constant.
  2. Find the length of \(B C\) in terms of \(k\). Given that the length of \(B C\) is 10 and using your answer to part (b),
  3. find the value of \(k\),
  4. find the coordinates of \(D\).
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 C1 2007 January Q4
14 marks Moderate -0.8
4 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 12 y + 12 = 0\).
  1. By completing the square, express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
  2. Write down:
    1. the coordinates of \(C\);
    2. the radius of the circle.
  3. Show that the circle does not intersect the \(x\)-axis.
  4. The line with equation \(x + y = 4\) intersects the circle at the points \(P\) and \(Q\).
    1. Show that the \(x\)-coordinates of \(P\) and \(Q\) satisfy the equation $$x ^ { 2 } + 3 x - 10 = 0$$
    2. Given that \(P\) has coordinates (2,2), find the coordinates of \(Q\).
    3. Hence find the coordinates of the midpoint of \(P Q\).