Area of region bounded by circle and line

7 \includegraphics[max width=\textwidth, alt={}, center]{01b98496-a717-4c68-8489-42d2203b700f-08_574_689_260_726} The diagram shows a sector \(A O B\) which is part of a circle with centre \(O\) and radius 6 cm and with angle \(A O B = 0.8\) radians. The point \(C\) on \(O B\) is such that \(A C\) is perpendicular to \(O B\). The arc \(C D\) is part of a circle with centre \(O\), where \(D\) lies on \(O A\). Find the area of the shaded region.

8 \includegraphics[max width=\textwidth, alt={}, center]{3bad1d9f-5b9e-4895-aa4e-3e6d9f6c072e-10_454_744_255_703} The diagram shows triangle \(A B C\) in which angle \(B\) is a right angle. The length of \(A B\) is 8 cm and the length of \(B C\) is 4 cm . The point \(D\) on \(A B\) is such that \(A D = 5 \mathrm {~cm}\). The sector \(D A C\) is part of a circle with centre \(D\).

1. Find the perimeter of the shaded region.
2. Find the area of the shaded region.

9 \includegraphics[max width=\textwidth, alt={}, center]{3b44e558-f91d-4175-acda-eceb70dad82c-12_497_652_260_744} In the diagram, arc \(A B\) is part of a circle with centre \(O\) and radius 8 cm . Arc \(B C\) is part of a circle with centre \(A\) and radius 12 cm , where \(A O C\) is a straight line.

1. Find angle \(B A O\) in radians.
2. Find the area of the shaded region.
3. Find the perimeter of the shaded region.

6 \includegraphics[max width=\textwidth, alt={}, center]{5e3e5418-7976-4232-8550-1da6420a3fcb-08_534_506_255_815} The diagram shows a motif formed by the major arc \(A B\) of a circle with radius \(r\) and centre \(O\), and the minor arc \(A O B\) of a circle, also with radius \(r\) but with centre \(C\). The point \(C\) lies on the circle with centre \(O\).

1. Given that angle \(A C B = k \pi\) radians, state the value of the fraction \(k\).
2. State the perimeter of the shaded motif in terms of \(\pi\) and \(r\).
3. Find the area of the shaded motif, giving your answer in terms of \(\pi , r\) and \(\sqrt { 3 }\).

10 \includegraphics[max width=\textwidth, alt={}, center]{88c7a3f3-e129-4e9c-acf8-8c96d2668d43-12_552_582_255_778} The diagram shows points \(A , B\) and \(C\) lying on a circle with centre \(O\) and radius \(r\). Angle \(A O B\) is 2.8 radians. The shaded region is bounded by two arcs. The upper arc is part of the circle with centre \(O\) and radius \(r\). The lower arc is part of a circle with centre \(C\) and radius \(R\).

1. State the size of angle \(A C O\) in radians.
2. Find \(R\) in terms of \(r\).
3. Find the area of the shaded region in terms of \(r\).

9 \includegraphics[max width=\textwidth, alt={}, center]{9803d51b-215e-4d03-884f-a67fb7ed6442-14_713_912_258_573} The diagram shows a circle with centre \(A\) and radius \(r\). Diameters CAD and BAE are perpendicular to each other. A larger circle has centre \(B\) and passes through \(C\) and \(D\).

1. Show that the radius of the larger circle is \(r \sqrt { 2 }\).
2. Find the area of the shaded region in terms of \(r\).

8 \includegraphics[max width=\textwidth, alt={}, center]{32a57386-2696-4fda-a3cb-ca0c5c3be432-3_600_883_1731_630} The diagram shows a rhombus \(A B C D\). Points \(P\) and \(Q\) lie on the diagonal \(A C\) such that \(B P D\) is an arc of a circle with centre \(C\) and \(B Q D\) is an arc of a circle with centre \(A\). Each side of the rhombus has length 5 cm and angle \(B A D = 1.2\) radians.

1. Find the area of the shaded region \(B P D Q\).
2. Find the length of \(P Q\).

7 \includegraphics[max width=\textwidth, alt={}, center]{a9e04003-1e43-40c4-991a-36aa3a93654b-3_718_899_258_621} The diagram shows a circle with centre \(A\) and radius \(r\). Diameters \(C A D\) and \(B A E\) are perpendicular to each other. A larger circle has centre \(B\) and passes through \(C\) and \(D\).

1. Show that the radius of the larger circle is \(r \sqrt { } 2\).
2. Find the area of the shaded region in terms of \(r\).

4 \includegraphics[max width=\textwidth, alt={}, center]{518bb805-5b14-4b41-94fd-38a31a90c218-06_401_698_255_721} The diagram shows a semicircle with centre \(O\) and radius 6 cm . The radius \(O C\) is perpendicular to the diameter \(A B\). The point \(D\) lies on \(A B\), and \(D C\) is an arc of a circle with centre \(B\).

1. Calculate the length of the \(\operatorname { arc } D C\).
2. Find the value of \(\frac { \text { area of region } P } { \text { area of region } Q }\),
   giving your answer correct to 3 significant figures.

7 \includegraphics[max width=\textwidth, alt={}, center]{097c5d00-9f92-4c3e-8056-7de09347fbb6-10_716_899_258_621} The diagram shows a circle with centre \(A\) and radius \(r\). Diameters \(C A D\) and \(B A E\) are perpendicular to each other. A larger circle has centre \(B\) and passes through \(C\) and \(D\).

1. Show that the radius of the larger circle is \(r \sqrt { } 2\).
2. Find the area of the shaded region in terms of \(r\).

7 \includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-05_848_1049_260_242} The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 9 y + 19 = 0\) and centre \(C\).

1. Find the following.
   The tangent to the circle at \(D\) meets the \(x\)-axis at the point \(A \left( \frac { 55 } { 4 } , 0 \right)\) and the \(y\)-axis at the point \(B ( 0 , - 11 )\).
2. Determine the area of triangle \(O B D\).

7. \begin{figure}[h] \end{figure} Figure 4 shows a circle with radius \(r _ { 1 }\) and a circle with radius \(r _ { 2 }\) The circles touch externally at a single point above the \(x\)-axis.
Both circles also have the \(x\)-axis as a tangent.

1. Show that the horizontal distance between the centres of the circles, \(d\), is given by $$d ^ { 2 } = 4 r _ { 1 } r _ { 2 }$$ The finite region \(R\), shown shaded in Figure 4, is bounded by the \(x\)-axis and minor arcs of the two circles. Given that \(r _ { 1 } \geqslant r _ { 2 }\)
2. show that the area of \(R\) is given by $$\left( r _ { 1 } + r _ { 2 } \right) \sqrt { r _ { 1 } r _ { 2 } } - \frac { 1 } { 2 } \left( r _ { 1 } ^ { 2 } - r _ { 2 } ^ { 2 } \right) \theta - \frac { 1 } { 2 } \pi r _ { 2 } ^ { 2 }$$ where \(\cos \theta = \frac { r _ { 1 } - r _ { 2 } } { r _ { 1 } + r _ { 2 } }\) Question 7 continues on the next page.
   
   \includegraphics[max width=\textwidth, alt={}]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-27_2269_53_306_36}
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-27_759_1378_269_347} \captionsetup{labelformat=empty} \caption{Figure 5}
   \end{figure} A sequence of circles, \(C _ { 1 } , C _ { 2 } , C _ { 3 } , \ldots\) with radii \(r _ { 1 } , r _ { 2 } , r _ { 3 } , \ldots\) respectively, is constructed such that
   • each circle is tangential to and above the \(x\)-axis
   • the first circle, \(C _ { 1 }\), has centre \(( 0,1 )\)
   • each successive circle touches the preceding one externally at a single point
   • the horizontal distances between the centres of successive circles form a geometric sequence with first term 2 and common ratio \(\frac { 1 } { \sqrt { 3 } }\)
   The first few circles in the sequence are shown in Figure 5.
3. 1. Determine the value of \(r _ { 3 }\)
   2. Show that, for \(n \geqslant 1 , r _ { n + 2 } = k r _ { n }\) where \(k\) is a constant to be determined.
   3. Hence show that, for \(n \geqslant 1 , r _ { 2 n } = r _ { 2 n - 1 }\) The region bounded between \(C _ { n } , C _ { n + 1 }\) and the \(x\)-axis is \(R _ { n }\) The total area, \(A\), bounded above the \(x\)-axis and under all the circles is the sum of the areas of all these regions.
4. Determine the value of \(A\), giving the answer in simplest form. \section*{Paper reference} \section*{Advanced Extension Award Mathematics} Insert for questions 5, 6 and 7
   Do not write on this insert.
   5. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-34_298_1040_212_516} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a sketch of a hexagon \(O A B C D E\) where
   Given that \(\overrightarrow { O A } = \mathbf { a }\) and \(\overrightarrow { O E } = \mathbf { e }\) determine as simplified expressions in terms of \(\mathbf { a }\) and \(\mathbf { e }\)
   1. \(\overrightarrow { A B }\)
   2. \(\overrightarrow { O D }\) The point \(R\) divides \(A B\) internally in the ratio \(1 : 2\)
   3. Determine \(\overrightarrow { R C }\) as a simplified expression in terms of \(\mathbf { a }\) and \(\mathbf { e }\) The line through the points \(R\) and \(C\) meets the line through the points \(O\) and \(D\) at the point \(X\).
   4. Determine \(\overrightarrow { O X }\) in the form \(\lambda \mathbf { a } + \mu \mathbf { e }\), where \(\lambda\) and \(\mu\) are real values in simplest form.
      6. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-35_236_1363_205_351} \captionsetup{labelformat=empty} \caption{Figure 3}
      \end{figure} Figure 3 shows a block \(A\) with mass \(4 m\) and a block \(B\) with mass \(5 m\).
      Block \(A\) is at rest on a rough plane inclined at an angle \(\alpha\) to the horizontal.
      Block \(B\) is at rest on a rough plane inclined at an angle \(\beta\) to the horizontal.
      The blocks are connected by a light inextensible string which passes over a small smooth pulley at the top of each plane. A small smooth ring \(C\), of mass \(8 m\), is threaded on the string between the pulleys so that \(A , B\) and \(C\) all lie in the same vertical plane. The part of the string between \(A\) and its pulley lies along a line of greatest slope of the plane of angle \(\alpha\). The part of the string between \(B\) and its pulley lies along a line of greatest slope of the plane of angle \(\beta\). The angle between the vertical and the string between each pulley and the ring \(C\) is \(\gamma\).
      The two blocks, \(A\) and \(B\), are modelled as particles.
      Given that
      7. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-36_721_1771_205_146} \captionsetup{labelformat=empty} \caption{Figure 4}
      \end{figure} Figure 4 shows a circle with radius \(r _ { 1 }\) and a circle with radius \(r _ { 2 }\) The circles touch externally at a single point above the \(x\)-axis.
      Both circles also have the \(x\)-axis as a tangent.
   5. Show that the horizontal distance between the centres of the circles, \(d\), is given by $$d ^ { 2 } = 4 r _ { 1 } r _ { 2 }$$ The finite region \(R\), shown shaded in Figure 4, is bounded by the \(x\)-axis and minor arcs of the two circles. Given that \(r _ { 1 } \geqslant r _ { 2 }\)
   6. show that the area of \(R\) is given by $$\left( r _ { 1 } + r _ { 2 } \right) \sqrt { r _ { 1 } r _ { 2 } } - \frac { 1 } { 2 } \left( r _ { 1 } ^ { 2 } - r _ { 2 } ^ { 2 } \right) \theta - \frac { 1 } { 2 } \pi r _ { 2 } ^ { 2 }$$ where \(\cos \theta = \frac { r _ { 1 } - r _ { 2 } } { r _ { 1 } + r _ { 2 } }\) Question 7 continues on the next page. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-37_761_1376_210_349} \captionsetup{labelformat=empty} \caption{Figure 5}
      \end{figure} A sequence of circles, \(C _ { 1 } , C _ { 2 } , C _ { 3 } , \ldots\) with radii \(r _ { 1 } , r _ { 2 } , r _ { 3 } , \ldots\) respectively, is constructed such that
      The first few circles in the sequence are shown in Figure 5.
   7. 1. Determine the value of \(r _ { 3 }\)
      2. Show that, for \(n \geqslant 1 , r _ { n + 2 } = k r _ { n }\) where \(k\) is a constant to be determined.
      3. Hence show that, for \(n \geqslant 1 , r _ { 2 n } = r _ { 2 n - 1 }\) The region bounded between \(C _ { n } , C _ { n + 1 }\) and the \(x\)-axis is \(R _ { n }\) The total area, \(A\), bounded above the \(x\)-axis and under all the circles is the sum of the areas of all these regions.
   8. Determine the value of \(A\), giving the answer in simplest form.

A circle \(C\) has centre \(M\) \((6, 4)\) and radius 3.

1. Write down the equation of the circle in the form $$(x - a)^2 + (y - b)^2 = r^2.$$ [2]

\includegraphics{figure_3} Figure 3 shows the circle \(C\). The point \(T\) lies on the circle and the tangent at \(T\) passes through the point \(P\) \((12, 6)\). The line \(MP\) cuts the circle at \(Q\).

1. Show that the angle \(TMQ\) is 1.0766 radians to 4 decimal places. [4]

The shaded region \(TPQ\) is bounded by the straight lines \(TP\), \(QP\) and the arc \(TQ\), as shown in Figure 3.

1. Find the area of the shaded region \(TPQ\). Give your answer to 3 decimal places. [5]

\includegraphics{figure_3} Figure 3 shows the circle \(C\) with equation $$x^2 + y^2 - 8x - 10y + 16 = 0.$$

1. Find the coordinates of the centre and the radius of \(C\). [3]

\(C\) crosses the \(y\)-axis at the points \(P\) and \(Q\).

1. Find the coordinates of \(P\) and \(Q\). [3]

The chord \(PQ\) subtends an angle of \(\theta\) at the centre of \(C\).

1. Using the cosine rule, show that \(\cos \theta = \frac{7}{25}\). [4]
2. Find the area of the shaded minor segment bounded by \(C\) and the chord \(PQ\). [4]

A circle, of radius \(\sqrt{5}\) and centre the origin \(O\), is divided into two segments by the line \(y = 1\).

1. Determine the area of the smaller segment. [4]

The line is rotated clockwise about \(O\) through \(45^{\circ}\) and then reflected in the \(x\)-axis.

1. Find the equation of the line in its final position. [3]