Circles

A circle has equation \((x - 2)^2 + (y + 3)^2 = 13\) Find the gradient of the tangent to this circle at the origin. Circle your answer. [1 mark] \(-\frac{3}{2}\) \quad \(-\frac{2}{3}\) \quad \(\frac{2}{3}\) \quad \(\frac{3}{2}\)

6 A circle has centre \(C ( - 3,1 )\) and radius \(\sqrt { 13 }\).

1. 1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
   2. Hence find the equation of the circle in the form $$x ^ { 2 } + y ^ { 2 } + m x + n y + p = 0$$ where \(m , n\) and \(p\) are integers.
2. The circle cuts the \(y\)-axis at the points \(A\) and \(B\). Find the distance \(A B\).
3. 1. Verify that the point \(D ( - 5 , - 2 )\) lies on the circle.
   2. Find the gradient of \(C D\).
   3. Hence find an equation of the tangent to the circle at the point \(D\).

14. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 16 y + k = 0$$ where \(k\) is a constant.

1. Find the coordinates of the centre of \(C\). Given that the radius of \(C\) is 10
2. find the value of \(k\). The point \(A ( a , - 16 )\), where \(a > 0\), lies on the circle \(C\). The tangent to \(C\) at the point \(A\) crosses the \(x\)-axis at the point \(D\) and crosses the \(y\)-axis at the point \(E\).
3. Find the exact area of triangle \(O D E\).

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 \(A B\).

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 \(x - 2 y - 12 = 0\).

Find the coordinates of the points of intersection of the circle \(x^2 + y^2 = 25\) and the line \(y = 3x\). Give your answers in surd form. [5]

A circle has centre \(C(3, -8)\) and radius \(10\).

1. Express the equation of the circle in the form $$(x - a)^2 + (y - b)^2 = k$$ [2 marks]
2. Find the \(x\)-coordinates of the points where the circle crosses the \(x\)-axis. [3 marks]
3. The line with equation \(y = 2x + 1\) intersects the circle.
   1. Show that the \(x\)-coordinates of the points of intersection satisfy the equation $$x^2 + 6x - 2 = 0$$ [3 marks]
   2. Hence show that the \(x\)-coordinates of the points of intersection are of the form \(m \pm \sqrt{n}\), where \(m\) and \(n\) are integers. [2 marks]

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.

In this question you must show detailed reasoning. A circle touches the lines \(y = \frac{1}{2}x\) and \(y = 2x\) at \((6, 3)\) and \((3, 6)\) respectively. \includegraphics{figure_6} Find the equation of the circle. [7]

7. \begin{figure}[h] \end{figure} The line \(y = 3 x - 4\) is a tangent to the circle \(C\), touching \(C\) at the point \(P ( 2,2 )\), as shown in Figure 1. The point \(Q\) is the centre of \(C\).

1. Find an equation of the straight line through \(P\) and \(Q\). Given that \(Q\) lies on the line \(y = 1\),
2. show that the \(x\)-coordinate of \(Q\) is 5,
3. find an equation for \(C\).

In this question you must show detailed reasoning. A circle touches the lines \(y = \frac{1}{2}x\) and \(y = 2x\) at \((6, 3)\) and \((3, 6)\) respectively. \includegraphics{figure_6} Find the equation of the circle. [7]

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. $$\begin{aligned} & ( x - 4 ) ^ { 2 } + ( y + 5 ) ^ { 2 } = 6 \\ & ( x + 4 ) ^ { 2 } + ( y - 5 ) ^ { 2 } = 6 \\ & ( x - 4 ) ^ { 2 } + ( y + 5 ) ^ { 2 } = 36 \\ & ( x + 4 ) ^ { 2 } + ( y - 5 ) ^ { 2 } = 36 \end{aligned}$$ □
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5 A circle \(S\) has centre at the point \(( 3,1 )\) and passes through the point \(( 0,5 )\).

1. Find the radius of \(S\) and hence write down its cartesian equation.
2. (a) Determine the two points on \(S\) where the \(y\)-coordinate is twice the \(x\)-coordinate.
   (b) Calculate the length of the minor arc joining these two points.

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 \(\left( \frac { 20 } { 3 } \pi + 8 \sqrt { 3 } \right) \mathrm { cm } ^ { 2 }\).

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]{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
   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.

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 \(B\).

10

1. The coordinates of two points \(A\) and \(B\) are \(( - 7,3 )\) and \(( 5,11 )\) respectively.
   Show that the equation of the perpendicular bisector of \(A B\) is \(3 x + 2 y = 11\).
2. A circle passes through \(A\) and \(B\) and its centre lies on the line \(12 x - 5 y = 70\). Find an equation of the circle.

\includegraphics{figure_4} A circle with centre \((9, -6)\) touches the \(x\)-axis as shown in Figure 4.

1. Write down an equation for the circle. [3] A line \(l\) is parallel to the \(x\)-axis. The line \(l\) cuts the circle at points \(P\) and \(Q\). Given that the distance \(PQ\) is 8
2. find the two possible equations for \(l\). [4]

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 \(O\).

\includegraphics{figure_1} The circle, with centre \(C\) and radius \(r\), touches the \(y\)-axis at \((0, 4)\) and also touches the line with equation \(4y - 3x = 0\), as shown in Fig. 1.

1. 1. Find the value of \(r\).
   2. Show that \(\arctan \left(\frac{4}{3}\right) + 2 \arctan \left(\frac{1}{2}\right) = \frac{1}{2} \pi\). [8]

The line with equation \(4x + 3y = q\), \(q > 12\), is a tangent to the circle.

1. Find the value of \(q\). [4]

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 \((x - 3)^2 + (y - 2)^2 = 20\) has centre C.

1. Write down the radius of the circle and the coordinates of C. [2]
2. Find the coordinates of the intersections of the circle with the \(x\)- and \(y\)-axes. [5]
3. Show that the points A\((1, 6)\) and B\((7, 4)\) lie on the circle. Find the coordinates of the midpoint of AB. Find also the distance of the chord AB from the centre of the circle. [5]

A circle has centre \(C\) which lies on the \(x\)-axis, as shown in the diagram. The line \(y = x\) meets the circle at \(A\) and \(B\). The midpoint of \(AB\) is \(M\). \includegraphics{figure_9} The equation of the circle is \(x^2 - 6x + y^2 + a = 0\), where \(a\) is a constant.

1. In this question you must show detailed reasoning. Find the \(x\)-coordinate of \(M\) and hence show that the area of triangle \(ABC\) is \(\frac{3}{2}\sqrt{9 - 2a}\). [6]
2. 1. Find the value of \(a\) when the area of triangle \(ABC\) is zero. [1]
   2. Give a geometrical interpretation of the case in part (b)(i). [1]
3. Give a geometrical interpretation of the case where \(a = 5\). [1]

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

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 \(L M N\) .

1. Show that \(x y = 2\)
2. 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,
3. 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)

In this question you must show detailed reasoning. A circle has equation \(x^2 + y^2 - 6x - 4y + 12 = 0\). Two tangents to this circle pass through the point \((0, 1)\). You are given that the scales on the \(x\)-axis and the \(y\)-axis are the same. Find the angle between these two tangents. [7]

5 A circle with centre \(C\) has equation \(( x + 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 25\).

1. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle.
2. 1. Verify that the point \(N ( 0 , - 2 )\) lies on the circle.
   2. Sketch the circle.
   3. Find an equation of the normal to the circle at the point \(N\).
3. The point \(P\) has coordinates (2, 6).
   1. Find the distance \(P C\), leaving your answer in surd form.
   2. Find the length of a tangent drawn from \(P\) to the circle.

In this question you must show detailed reasoning. A circle has equation \(x^2 + y^2 - 6x - 4y + 12 = 0\). Two tangents to this circle pass through the point \((0, 1)\). You are given that the scales on the \(x\)-axis and the \(y\)-axis are the same. Find the angle between these two tangents. [7]

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.

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 \(B\).

6. \begin{figure}[h] \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.

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.

1. The circle \(C _ { 1 }\) has equation

$$x ^ { 2 } + y ^ { 2 } + 8 x - 10 y = 29$$

1. 1. Find the coordinates of the centre of \(C _ { 1 }\)
   2. Find the exact value of the radius of \(C _ { 1 }\) In part (b) you must show detailed reasoning.
      The circle \(C _ { 2 }\) has equation $$( x - 5 ) ^ { 2 } + ( y + 8 ) ^ { 2 } = 52$$
2. Prove that the circles \(C _ { 1 }\) and \(C _ { 2 }\) neither touch nor intersect.

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.

The ellipse \(E\) has equation $$x^2 + \frac{y^2}{9} = 1$$ The line with equation \(y = mx + 4\) is a tangent to \(E\) Without using differentiation show that \(m = \pm\sqrt{7}\) [4 marks]

5 A circle has equation \(( x - 2 ) ^ { 2 } + y ^ { 2 } = 20\).

1. Write down the radius of the circle and the coordinates of its centre.
2. Find the points of intersection of the circle with the \(y\)-axis and sketch the circle.
3. Show that, where the line \(y = 2 x + k\) intersects the circle, $$5 x ^ { 2 } + ( 4 k - 4 ) x + k ^ { 2 } - 16 = 0$$
4. Hence find the values of \(k\) for which the line \(y = 2 x + k\) is a tangent to the circle.

The ellipse \(E\) has equation $$x^2 + \frac{y^2}{9} = 1$$ The line with equation \(y = mx + 4\) is a tangent to \(E\) Without using differentiation show that \(m = \pm\sqrt{7}\) [4 marks]

The equation of a circle is \((x - 3)^2 + y^2 = 18\). The line with equation \(y = mx + c\) passes through the point \((0, -9)\) and is a tangent to the circle. Find the two possible values of \(m\) and, for each value of \(m\), find the coordinates of the point at which the tangent touches the circle. [8]

8. \begin{figure}[h] \end{figure} Figure 2 shows a circle \(C\) with centre \(O\) and radius 5

1. Write down the cartesian equation of \(C\). The points \(P ( - 3 , - 4 )\) and \(Q ( 3 , - 4 )\) lie on \(C\).
2. Show that the tangent to \(C\) at the point \(Q\) has equation $$3 x - 4 y = 25$$
3. Show that, to 3 decimal places, angle \(P O Q\) is 1.287 radians. The tangent to \(C\) at \(P\) and the tangent to \(C\) at \(Q\) intersect on the \(y\)-axis at the point \(R\).
4. Find the area of the shaded region \(P Q R\) shown in Figure 2. \includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-25_177_154_2576_1804}

4.The line with equation \(y = m x\) is a tangent to the circle \(C _ { 1 }\) with equation $$( x + 4 ) ^ { 2 } + ( y - 7 ) ^ { 2 } = 13$$

1. 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 \(B\) .
2. 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 \(Q\) .
3. Find the coordinates of either the point \(P\) or the point \(Q\) .
   (2)

In this question you must show detailed reasoning. The centre of a circle C is 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]

11. \begin{figure}[h] \end{figure} Figure 3 shows the circle \(C\) with equation $$x ^ { 2 } + y ^ { 2 } - 10 x - 8 y + 32 = 0$$ and the line \(l\) with equation $$2 y + x + 6 = 0$$

1. Find
   1. the coordinates of the centre of \(C\),
   2. the radius of \(C\).
2. Find the shortest distance between \(C\) and \(l\).

8. The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 2 x - 18 y + 73 = 0\) and the straight line with equation \(y = 2 x - 3\).

1. Find the coordinates of the centre and the radius of the circle.
2. Find the coordinates of the point on the line which is closest to 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 \(k\).

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 circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } + 6 k x - 2 k y + 7 = 0$$ where \(k\) is a constant.

1. Find in terms of \(k\),
   1. the coordinates of the centre of \(C\)
   2. the radius of \(C\) The line with equation \(y = 2 x - 1\) intersects \(C\) at 2 distinct points.
2. Find the range of possible values of \(k\).

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.

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 \(C\).

9 The line \(y = 2 x + 5\) intersects the circle with equation \(x ^ { 2 } + y ^ { 2 } = 20\) at \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\) in surd form and hence find the exact length of the chord \(A B\).
   A straight line through the point \(( 10,0 )\) with gradient \(m\) is a tangent to the circle.
2. Find the two possible values of \(m\).

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } + 8 x - 4 y = 0$$

1. Find
   1. the coordinates of the centre of \(C\),
   2. the exact radius of \(C\). The point \(P\) lies on \(C\).
      Given that the tangent to \(C\) at \(P\) has equation \(x + 2 y + 10 = 0\)
2. find the coordinates of \(P\)
3. Find the equation of the normal to \(C\) at \(P\), giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are integers to be found.

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.