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

10 The circle \(x ^ { 2 } + y ^ { 2 } + 4 x - 2 y - 20 = 0\) has centre \(C\) and passes through points \(A\) and \(B\).

1. State the coordinates of \(C\).
   It is given that the midpoint, \(D\), of \(A B\) has coordinates \(\left( 1 \frac { 1 } { 2 } , 1 \frac { 1 } { 2 } \right)\).
2. Find the equation of \(A B\), giving your answer in the form \(y = m x + c\).
3. Find, by calculation, the \(x\)-coordinates of \(A\) and \(B\).

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.

7 The point \(A\) has coordinates ( \(- 1,6\) ) and the point \(B\) has coordinates (7,2).

1. Find the equation of the perpendicular bisector of \(A B\), giving your answer in the form \(y = m x + c\).
2. A point \(C\) on the perpendicular bisector has coordinates \(( p , q )\). The distance \(O C\) is 2 units, where \(O\) is the origin. Write down two equations involving \(p\) and \(q\) and hence find the coordinates of the possible positions of \(C\).

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 \(r\).

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 \(r\).

4. Use algebra to solve the simultaneous equations $$\begin{array} { r } y - 3 x = 4 \\ x ^ { 2 } + y ^ { 2 } + 6 x - 4 y = 4 \end{array}$$ You must show all stages of your working.

15. \begin{figure}[h] \end{figure} Diagram NOT drawn to scale The points \(X\) and \(Y\) have coordinates \(( 0,3 )\) and \(( 6,11 )\) respectively. \(X Y\) is a chord of a circle \(C\) with centre \(Z\), as shown in Figure 3.

1. Find the gradient of \(X Y\). The point \(M\) is the midpoint of \(X Y\).
2. Find an equation for the line which passes through \(Z\) and \(M\). Given that the \(y\) coordinate of \(Z\) is 10 ,
3. find the \(x\) coordinate of \(Z\),
4. find the equation of the circle \(C\), giving your answer in the form $$x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0$$ where \(a\), \(b\) and \(c\) are constants.

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } + 10 x - 6 y + 18 = 0$$ Find

1. the coordinates of the centre of \(C\),
2. the radius of \(C\). The circle \(C\) meets the line with equation \(x = - 3\) at two points.
3. Find the exact values for the \(y\) coordinates of these two points, giving your answers as fully simplified surds.

15. The points \(A\) and \(B\) have coordinates \(( - 8 , - 8 )\) and \(( 12,2 )\) respectively. \(A B\) is the diameter of a circle \(C\).

1. Find an equation for the circle \(C\). The point \(( 4,8 )\) also lies on \(C\).
2. Find an equation of the tangent to \(C\) at the point ( 4,8 ), giving your answer in the form \(a x + b y + c = 0\)

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.

9. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 10 x - 6 y + 9 = 0$$

1. Find the coordinates of the centre of \(C\).
2. Find the radius of \(C\). The point \(P ( - 2,7 )\) lies on \(C\).
3. Find an equation of the tangent to \(C\) at the point \(P\). Write your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
   

11. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the circle \(C\) with centre \(Q\) and equation $$x ^ { 2 } + y ^ { 2 } - 6 x + 2 y + 5 = 0$$

1. Find
   1. the coordinates of \(Q\),
   2. the exact value of the radius of \(C\). The tangents to \(C\) from the point \(T ( 8,4 )\) meet \(C\) at the points \(M\) and \(N\), as shown in Figure 4.
2. Show that the obtuse angle \(M Q N\) is 2.498 radians to 3 decimal places. The region \(R\), shown shaded in Figure 4, is bounded by the tangent \(T N\), the minor arc \(N M\), and the tangent \(M T\).
3. Find the area of region \(R\).

15. \begin{figure}[h] \end{figure} Diagram not drawn to scale The circle shown in Figure 4 has centre \(P ( 5,6 )\) and passes through the point \(A ( 12,7 )\). Find

1. the exact radius of the circle,
2. an equation of the circle,
3. an equation of the tangent to the circle at the point \(A\). The circle also passes through the points \(B ( 0,1 )\) and \(C ( 4,13 )\).
4. Use the cosine rule on triangle \(A B C\) to find the size of the angle \(B C A\), giving your answer in degrees to 3 significant figures.

12. \begin{figure}[h] \end{figure} Figure 3 shows a circle \(C\) \(C\) touches the \(y\)-axis and has centre at the point ( \(a , 0\) ) where \(a\) is a positive constant.

1. Write down an equation for \(C\) in terms of \(a\) Given that the point \(P ( 4 , - 3 )\) lies on \(C\),
2. find the value of \(a\)

14. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the circle \(C _ { 1 }\) The points \(A ( 1,4 )\) and \(B ( 7,8 )\) lie on \(C _ { 1 }\) Given that \(A B\) is a diameter of the circle \(C _ { 1 }\)

1. find the coordinates for the centre of \(C _ { 1 }\)
2. find the exact radius of \(C _ { 1 }\) simplifying your answer. Two distinct circles \(C _ { 2 }\) and \(C _ { 3 }\) each have centre \(( 0,0 )\).
   Given that each of these circles touch circle \(C _ { 1 }\)
3. find the equation of circle \(C _ { 2 }\) and the equation of circle \(C _ { 3 }\)

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 \(B\).

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } + 4 x + p y + 123 = 0$$ where \(p\) is a constant. Given that the point \(( 1,16 )\) lies on \(C\),

1. find
   1. the value of \(p\),
   2. the coordinates of the centre of \(C\),
   3. the radius of \(C\).
2. Find an equation of the tangent to \(C\) at the point ( 1,16 ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found. \includegraphics[max width=\textwidth, alt={}, center]{de511cb3-35c7-4225-b459-a136b6304b78-31_33_19_2668_1896}

13. The circle \(C\) has centre \(A ( 1 , - 3 )\) and passes through the point \(P ( 8 , - 2 )\).

1. Find an equation for the circle \(C\). The line \(l _ { 1 }\) is the tangent to \(C\) at the point \(P\).
2. Find an equation for \(l _ { 1 }\), giving your answer in the form \(y = m x + c\) The line \(l _ { 2 }\), with equation \(y = x + 6\), is the tangent to \(C\) at the point \(Q\).
3. Find the coordinates of the point \(Q\).

1. The circle \(C\) has equation

$$( x - 3 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 30$$ Write down

1. 1. the coordinates of the centre of \(C\),
   2. the exact value of the radius of \(C\). Given that the point \(P\) with coordinates \(( 6 , k )\), where \(k\) is a constant, lies inside circle \(C\), (b) show that $$k ^ { 2 } + 8 k - 5 < 0$$
2. Hence find the exact set of values of \(k\) for which \(P\) lies inside \(C\). \includegraphics[max width=\textwidth, alt={}, center]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-34_2256_52_315_1978}

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 \(O D E\).

12. The circle \(C\) has centre \(A ( 2,1 )\) and passes through the point \(B ( 10,7 )\)

1. Find an equation for \(C\). The line \(l _ { 1 }\) is the tangent to \(C\) at the point \(B\).
2. Find an equation for \(l _ { 1 }\) The line \(l _ { 2 }\) is parallel to \(l _ { 1 }\) and passes through the mid-point of \(A B\).
   Given that \(l _ { 2 }\) intersects \(C\) at the points \(P\) and \(Q\),
3. find the length of \(P Q\), giving your answer in its simplest surd form.

6. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + 6 x - 4 y - 14 = 0$$

1. Find
   1. the coordinates of the centre of \(C\),
   2. the exact radius of \(C\). The line with equation \(y = k\), where \(k\) is a constant, is a tangent to \(C\).
2. Find the possible values of \(k\). The line with equation \(y = p\), where \(p\) is a negative constant, is a chord of \(C\).
   Given that the length of this chord is 4 units,
3. find the value of \(p\).
   

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9. A circle \(C\) has equation $$( x - k ) ^ { 2 } + ( y - 2 k ) ^ { 2 } = k + 7$$ where \(k\) is a positive constant.

1. Write down, in terms of \(k\),
   1. the coordinates of the centre of \(C\),
   2. the radius of \(C\). Given that the point \(P ( 2,3 )\) lies on \(C\)
2. 1. show that \(5 k ^ { 2 } - 17 k + 6 = 0\)
   2. hence find the possible values of \(k\). The tangent to the circle at \(P\) intersects the \(x\)-axis at point \(T\).
      Given that \(k < 2\)
3. calculate the exact area of triangle \(O P T\).

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.