Circles

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 \(B\).

1 Line \(L\) has equation $$5 y = 4 x + 6$$ Find the gradient of a line parallel to line \(L\)
Circle your answer.
\(- \frac { 5 } { 4 }\)
\(- \frac { 4 } { 5 }\)
\(\frac { 4 } { 5 }\)
\(\frac { 5 } { 4 }\)

11. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 8 x - 10 y + 16 = 0$$ The centre of \(C\) is at the point \(T\).

1. Find
   1. the coordinates of the point \(T\),
   2. the radius of the circle \(C\). The point \(M\) has coordinates \(( 20,12 )\).
2. Find the exact length of the line \(M T\). Point \(P\) lies on the circle \(C\) such that the tangent at \(P\) passes through the point \(M\).
3. Find the exact area of triangle \(M T P\), giving your answer as a simplified surd.

3. The line joining the points \(( - 1,4 )\) and \(( 3,6 )\) is a diameter of the circle \(C\). Find an equation for \(C\).

3. \begin{figure}[h] \end{figure} 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 \(A B\),
2. the coordinates of the midpoint \(P\) of \(A B\),
3. an equation for the circle \(C\).

5 A circle has equation \(( x - 1 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 40\). A line with equation \(y = x - 9\) intersects the circle at points \(A\) and \(B\).

1. Find the coordinates of the two points of intersection.
2. Find an equation of the circle with diameter \(A B\).

1. Find, by calculation, the coordinates of \(A\) and \(B\).
2. Find an equation of the circle which has its centre at \(C\) and for which the line with equation \(y = 3 x - 20\) is a tangent to the circle.

8 In this question you must show detailed reasoning. The centre of a circle C is at the point \(( - 1,3 )\) and C passes through the point \(( 1 , - 1 )\). The straight line L passes through the points \(( 1,9 )\) and \(( 4,3 )\). Show that L is a tangent to C .

7．A circle \(C\) has centre \(X ( a , b )\) and radius \(r\) ．
A line \(l\) has equation \(y = m x + c\)
（a）Show that the \(x\) coordinates of the points where \(C\) and \(l\) intersect satisfy $$\left( m ^ { 2 } + 1 \right) x ^ { 2 } - 2 ( a - m ( c - b ) ) x + a ^ { 2 } + ( c - b ) ^ { 2 } - r ^ { 2 } = 0$$ Given that \(l\) is a tangent to \(C\) ，
（b）show that $$c = b - m a \pm r \sqrt { m ^ { 2 } + 1 }$$ The circle \(C _ { 1 }\) has equation $$x ^ { 2 } + y ^ { 2 } - 16 = 0$$ and the circle \(C _ { 2 }\) has equation $$x ^ { 2 } + y ^ { 2 } - 20 x - 10 y + 89 = 0$$ （c）Find the equations of any lines that are normal to both \(C _ { 1 }\) and \(C _ { 2 }\) ，justifying your answer．
（d）Find the equations of all lines that are a tangent to both \(C _ { 1 }\) and \(C _ { 2 }\)
［You may find the following Pythagorean triple helpful in this part： \(7 ^ { 2 } + 24 ^ { 2 } = 25 ^ { 2 }\) ］

4 The circle \(x ^ { 2 } + y ^ { 2 } - 6 x + 4 y + k = 0\) has radius 5.
Determine the value of \(k\).

3 A circle has equation $$( x - 5 ) ^ { 2 } + ( y - 13 ) ^ { 2 } = 16$$ Find the radius of the circle. Circle your answer. 41216256

9 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 2 y - 3 = 0\).

1. Find the coordinates of \(C\) and the radius of the circle.
2. Find the values of \(k\) for which the line \(y = k\) is a tangent to the circle, giving your answers in simplified surd form.
3. The points \(S\) and \(T\) lie on the circumference of the circle. \(M\) is the mid-point of the chord \(S T\). Given that the length of \(C M\) is 2 , calculate the length of the chord \(S T\).
4. Find the coordinates of the point where the circle meets the line \(x - 2 y - 12 = 0\).

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.

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 \(x ^ { 2 } + y ^ { 2 } - 3 y - 4 = 0\).

8 In this question you must show detailed reasoning. The lines \(y = \frac { 1 } { 2 } x\) and \(y = - \frac { 1 } { 2 } x\) are tangents to a circle at \(( 2,1 )\) and \(( - 2,1 )\) respectively. Find the equation of the circle in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\), where \(a , b\) and \(c\) are constants.

10 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } - 4 x + 6 y - 77 = 0\).

1. Find the \(x\)-coordinates of the points \(A\) and \(B\) where the circle intersects the \(x\)-axis.
2. Find the point of intersection of the tangents to the circle at \(A\) and \(B\).

10 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } - 4 x + 6 y - 77 = 0\).

1. Find the \(x\)-coordinates of the points \(A\) and \(B\) where the circle intersects the \(x\)-axis.
2. Find the point of intersection of the tangents to the circle at \(A\) and \(B\).

4．The line with equation \(y = m x\) is a tangent to the circle \(C _ { 1 }\) with equation $$( x + 4 ) ^ { 2 } + ( y - 7 ) ^ { 2 } = 13$$ （a）Show that \(m\) satisfies the equation $$3 m ^ { 2 } + 56 m + 36 = 0$$ The tangents from the origin \(O\) to \(C _ { 1 }\) touch \(C _ { 1 }\) at the points \(A\) and \(B\) ．
（b）Find the coordinates of the points \(A\) and \(B\) ．
（8）
Another circle \(C _ { 2 }\) has equation \(x ^ { 2 } + y ^ { 2 } = 13\) ．The tangents from the point \(( 4 , - 7 )\) to \(C _ { 2 }\) touch it at the points \(P\) and \(Q\) ．
（c）Find the coordinates of either the point \(P\) or the point \(Q\) ．
（2）

5 Points \(A ( 7,12 )\) and \(B\) lie on a circle with centre \(( - 2,5 )\). The line \(A B\) has equation \(y = - 2 x + 26\).
Find the coordinates of \(B\).

13. The point \(A ( 9 , - 13 )\) lies on a circle \(C\) with centre the origin and radius \(r\).

1. Find the exact value of \(r\).
2. Find an equation of the circle \(C\). A straight line through point \(A\) has equation \(2 y + 3 x = k\), where \(k\) is a constant.
3. Find the value of \(k\). This straight line cuts the circle again at the point \(B\).
4. Find the exact coordinates of point \(B\).

12
\includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-18_1006_938_269_591} The diagram shows a circle \(P\) with centre \(( 0,2 )\) and radius 10 and the tangent to the circle at the point \(A\) with coordinates \(( 6,10 )\). It also shows a second circle \(Q\) with centre at the point where this tangent meets the \(y\)-axis and with radius \(\frac { 5 } { 2 } \sqrt { 5 }\).

1. Write down the equation of circle \(P\).
2. Find the equation of the tangent to the circle \(P\) at \(A\).
3. Find the equation of circle \(Q\) and hence verify that the \(y\)-coordinates of both of the points of intersection of the two circles are 11.
4. Find the coordinates of the points of intersection of the tangent and circle \(Q\), giving the answers in surd form.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 A sector of a circle of radius \(r \mathrm {~cm}\) has an area of \(A \mathrm {~cm} ^ { 2 }\). Express the perimeter of the sector in terms of \(r\) and \(A\).

4 A circle has diameter \(d\), circumference \(C\), and area \(A\). Starting with the standard formulae for a circle, show that \(C d = k A\), finding the numerical value of \(k\).

8
\includegraphics[max width=\textwidth, alt={}, center]{80a20f05-61db-42d9-b4ba-53eea2290b2d-10_780_814_264_662} The diagram shows a symmetrical metal plate. The plate is made by removing two identical pieces from a circular disc with centre \(C\). The boundary of the plate consists of two \(\operatorname { arcs } P S\) and \(Q R\) of the original circle and two semicircles with \(P Q\) and \(R S\) as diameters. The radius of the circle with centre \(C\) is 4 cm , and \(P Q = R S = 4 \mathrm {~cm}\) also.

1. Show that angle \(P C S = \frac { 2 } { 3 } \pi\) radians.
2. Find the exact perimeter of the plate.
3. Show that the area of the plate is \(\left( \frac { 20 } { 3 } \pi + 8 \sqrt { 3 } \right) \mathrm { cm } ^ { 2 }\).

8. The circle \(C\) has radius 5 and its centre is the origin. The point \(T\) has coordinates \(( 11,0 )\).
The tangents from \(T\) to the circle \(C\) touch \(C\) at the points \(R\) and \(S\).

1. Write down the geometrical name for the quadrilateral ORTS.
2. Find the exact value of the area of the quadrilateral ORTS. Give your answer in its simplest form.

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.

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
   • the interior angle at \(O\) and at \(C\) are each \(60 ^ { \circ }\)
   • the interior angle at each of the other vertices is \(150 ^ { \circ }\)
   • \(O A = O E = B C = C D\)
   • \(A B = E D = 3 \times O A\)
   Given that \(\overrightarrow { O A } = \mathbf { a }\) and \(\overrightarrow { O E } = \mathbf { e }\)
   determine as simplified expressions in terms of \(\mathbf { a }\) and \(\mathbf { e }\)
5. \(\overrightarrow { A B }\)
6. \(\overrightarrow { O D }\) The point \(R\) divides \(A B\) internally in the ratio \(1 : 2\)
7. 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 \(X\).
8. 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
   • \(\tan \alpha = \frac { 5 } { 12 }\) and \(\tan \beta = \frac { 7 } { 24 }\) and \(\tan \gamma = \frac { 3 } { 4 }\)
   • the coefficient of friction, \(\mu\), is the same between each block and its plane
   • one of the blocks is on the point of sliding up its plane
   • the tension in the string is \(T\)
   • determine, in terms of \(m\) and \(g\), an expression for \(T\),
   • draw a diagram showing the forces on block \(A\), clearly labelling each of the forces acting on the block,
   • determine the value of \(\mu\), giving a justification for your answer.
   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.
9. 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 \(r _ { 1 } \geqslant r _ { 2 }\)
10. 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
    • 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.
11. 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.
12. Determine the value of \(A\), giving the answer in simplest form.

15 The circle \(x ^ { 2 } + y ^ { 2 } + 2 x - 14 y + 25 = 0\) has its centre at the point \(C\). The line \(7 y = x + 25\) intersects the circle at points A and B . Prove that triangle ABC is a right-angled triangle.

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.

4. \begin{figure}[h] \end{figure} Figure 1 shows the equilateral triangle \(L M N\) of side 2 cm ．The point \(P\) lies on \(L M\) such that \(L P = x \mathrm {~cm}\) and the point \(Q\) lies on \(L N\) such that \(L Q = y \mathrm {~cm}\) ．The points \(P\) and \(Q\) are chosen so that the area of triangle \(L P Q\) is half the area of triangle \(L M N\) ．
（a）Show that \(x y = 2\)
（b）Find the shortest possible length of \(P Q\) ，justifying your answer． Mathematicians know that for any closed curve or polygon enclosing a fixed area，the ratio \(\frac { \text { area enclosed } } { \text { perimeter } }\) is a maximum when the closed curve is a circle． By considering 6 copies of triangle \(L M N\) suitably arranged，
（c）find the length of the shortest line or curve that can be drawn from a point on \(L M\) to a point on \(L N\) to divide the area of triangle \(L M N\) in half．Justify your answer．
（6）

7 A circle with centre \(( 5,2 )\) passes through the point \(( 7,5 )\).

1. Find an equation of the circle.
   The line \(y = 5 x - 10\) intersects the circle at \(A\) and \(B\).
2. Find the exact length of the chord \(A B\).

9. The circle \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 12 x + 8 y + 16 = 0$$

1. Find the coordinates of the centre of \(C\).
2. Find the radius of \(C\).
3. Sketch \(C\). Given that \(C\) crosses the \(x\)-axis at the points \(A\) and \(B\),
4. find the length \(A B\), giving your answer in the form \(k \sqrt { 5 }\).

7 A circle with centre \(( 5,2 )\) passes through the point \(( 7,5 )\).

1. Find an equation of the circle.
   The line \(y = 5 x - 10\) intersects the circle at \(A\) and \(B\).
2. Find the exact length of the chord \(A B\).

The circle \(C\) has centre \(( 3,4 )\) and passes through the point \(( 8 , - 8 )\). Find an equation for C

1 A circle has centre \(( 4 , - 5 )\) and radius 6
Find the equation of the circle.
Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & ( x - 4 ) ^ { 2 } + ( y + 5 ) ^ { 2 } = 6 \\ & ( x + 4 ) ^ { 2 } + ( y - 5 ) ^ { 2 } = 6 \\ & ( x - 4 ) ^ { 2 } + ( y + 5 ) ^ { 2 } = 36 \\ & ( x + 4 ) ^ { 2 } + ( y - 5 ) ^ { 2 } = 36 \end{aligned}$$ □
□
□
□
□

6 The vertices of triangle \(A B C\) are \(A ( - 3,1 ) , B ( 5,0 )\) and \(C ( 9,7 )\).

1. Show that \(A B = B C\).
2. Show that angle \(A B C\) is not a right angle.
3. Find the coordinates of the midpoint of \(A C\).
4. Determine the equation of the line of symmetry of the triangle, giving your answer in the form \(p x + q y = r\), where \(p , q\) and \(r\) are integers to be determined.
5. Write down an equation of the circle with centre \(A\) which passes through \(B\). This circle intersects the line of symmetry of the triangle at \(B\) and at a second point.
6. Find the coordinates of this second point.

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.

6 Circles \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$x ^ { 2 } + y ^ { 2 } + 6 x - 10 y + 18 = 0 \text { and } ( x - 9 ) ^ { 2 } + ( y + 4 ) ^ { 2 } - 64 = 0$$ respectively.

1. Find the distance between the centres of the circles.
   \(P\) and \(Q\) are points on \(C _ { 1 }\) and \(C _ { 2 }\) respectively. The distance between \(P\) and \(Q\) is denoted by \(d\).
2. Find the greatest and least possible values of \(d\).

11. \begin{figure}[h] \end{figure} Circle \(C _ { 1 }\) has equation \(x ^ { 2 } + y ^ { 2 } = 100\)
Circle \(C _ { 2 }\) has equation \(( x - 15 ) ^ { 2 } + y ^ { 2 } = 40\)
The circles meet at points \(A\) and \(B\) as shown in Figure 3.

1. Show that angle \(A O B = 0.635\) radians to 3 significant figures, where \(O\) is the origin. The region shown shaded in Figure 3 is bounded by \(C _ { 1 }\) and \(C _ { 2 }\)
2. Find the perimeter of the shaded region, giving your answer to one decimal place.

1. Express \(\sqrt { 22.5 }\) in the form \(k \sqrt { 10 }\).
2. A circle has the equation

$$x ^ { 2 } + y ^ { 2 } + 8 x - 4 y + k = 0$$ where \(k\) is a constant.

1. Find the coordinates of the centre of the circle. Given that the \(x\)-axis is a tangent to the circle,
2. find the value of \(k\).

5 A circle with centre \(C ( - 5,6 )\) touches the \(y\)-axis, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{66813123-3876-4484-aad1-4bfc09bb1508-6_444_698_372_680}

1. Find the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
2. 1. Verify that the point \(P ( - 2,2 )\) lies on the circle.
   2. Find an equation of the normal to the circle at the point \(P\).
   3. The mid-point of \(P C\) is \(M\). Determine whether the point \(P\) is closer to the point \(M\) or to the origin \(O\).

3. $$\mathrm { f } ( x ) = x ^ { 3 } - ( k + 4 ) x + 2 k , \quad \text { where } k \text { is a constant. }$$ （a）Show that，for all values of \(k\) ，the curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 2,0 )\) ．
（b）Find the values of \(k\) for which the equation \(\mathrm { f } ( x ) = 0\) has exactly two distinct roots． Given that \(k > 0\) ，that the \(x\)－axis is a tangent to the curve with equation \(y = \mathrm { f } ( x )\) ，and that the line \(y = p\) intersects the curve in three distinct points，
（c）find the set of values that \(p\) can take．
\includegraphics[max width=\textwidth, alt={}, center]{a243ceda-8175-4ae0-9bc7-b3048f468d10-3_573_899_343_704} The circle, with centre \(C\) and radius \(r\), touches the \(y\)-axis at \(( 0,4 )\) and also touches the line with equation \(4 y - 3 x = 0\), as shown in Fig. 1.

1. 1. Find the value of \(r\).
   2. Show that \(\arctan \left( \frac { 3 } { 4 } \right) + 2 \arctan \left( \frac { 1 } { 2 } \right) = \frac { 1 } { 2 } \pi\).
      (8) The line with equation \(4 x + 3 y = q , q > 12\), is a tangent to the circle.
2. Find the value of \(q\).
   (4)

1 Points \(A\) and \(B\) have coordinates \(( 5,2 )\) and \(( 10 , - 1 )\) respectively.

1. Find the equation of the perpendicular bisector of \(A B\).
2. Find the equation of the circle with centre \(A\) which passes through \(B\).

1 Points \(A\) and \(B\) have coordinates \(( 5,2 )\) and \(( 10 , - 1 )\) respectively.

1. Find the equation of the perpendicular bisector of \(A B\).
2. Find the equation of the circle with centre \(A\) which passes through \(B\).

11 \begin{figure}[h] \end{figure} Fig. 11 shows the line through the points \(\mathrm { A } ( - 1,3 )\) and \(\mathrm { B } ( 5,1 )\).

1. Find the equation of the line through A and B .
2. Show that the area of the triangle bounded by the axes and the line through A and B is \(\frac { 32 } { 3 }\) square units.
3. Show that the equation of the perpendicular bisector of AB is \(y = 3 x - 4\).
4. A circle passing through A and B has its centre on the line \(x = 3\). Find the centre of the circle and hence find the radius and equation of the circle.

3. A circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 4 x + 10 y = k$$ where \(k\) is a constant.

1. Find the coordinates of the centre of \(C\).
2. State the range of possible values for \(k\).

1．A point \(P\) lies on the curve with equation $$x ^ { 2 } + y ^ { 2 } - 6 x + 8 y = 24$$ Find the greatest and least possible values of the length \(O P\) ，where \(O\) is the origin．

6 In this question you must show detailed reasoning. 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.

9
\includegraphics[max width=\textwidth, alt={}, center]{73c0c113-8f35-4e7f-ad5d-604602498b71-4_837_1020_255_559} The diagram shows two circles, \(C _ { 1 }\) and \(C _ { 2 }\), touching at the point \(T\). Circle \(C _ { 1 }\) has centre \(P\) and radius 8 cm ; circle \(C _ { 2 }\) has centre \(Q\) and radius 2 cm . Points \(R\) and \(S\) lie on \(C _ { 1 }\) and \(C _ { 2 }\) respectively, and \(R S\) is a tangent to both circles.

1. Show that \(R S = 8 \mathrm {~cm}\).
2. Find angle \(R P Q\) in radians correct to 4 significant figures.
3. Find the area of the shaded region.

11

1. Show that the equation of the radius of the circle through \(P\) is \(y = 7 x - 37\)
   

   11	Find the equation of \(C\)		Do not write outside the box
   		[4 marks]