1.03e Complete the square: find centre and radius of circle

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.

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 \(B\).

9 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 26 = 0\).

1. Find the coordinates of the centre of the circle and the radius. Hence find the coordinates of the lowest point on the circle.
2. Find the set of values of the constant \(k\) for which the line with equation \(y = k x - 5\) intersects the circle at two distinct points.

8 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } + a x + b y - 12 = 0\). The points \(A ( 1,1 )\) and \(B ( 2 , - 6 )\) lie on the circle.

1. Find the values of \(a\) and \(b\) and hence find the coordinates of the centre of the circle.
2. Find the equation of the tangent to the circle at the point \(A\), giving your answer in the form \(p x + q y = k\), where \(p , q\) and \(k\) are integers.

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.

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 \(A B\).

8 A circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 2 y - 15 = 0\) meets the \(y\)-axis at the points \(A\) and \(B\). The tangents to the circle at \(A\) and \(B\) meet at the point \(P\). Find the coordinates of \(P\). \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-10_71_1659_466_244} \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-10_2723_37_136_2010}

12 A diameter of a circle \(C _ { 1 }\) has end-points at \(( - 3 , - 5 )\) and \(( 7,3 )\).

1. Find an equation of the circle \(C _ { 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{01b98496-a717-4c68-8489-42d2203b700f-16_618_846_1062_644} The circle \(C _ { 1 }\) is translated by \(\binom { 8 } { 4 }\) to give circle \(C _ { 2 }\), as shown in the diagram.
2. Find an equation of the circle \(C _ { 2 }\).
   The two circles intersect at points \(R\) and \(S\).
3. Show that the equation of the line \(R S\) is \(y = - 2 x + 13\).
4. Hence show that the \(x\)-coordinates of \(R\) and \(S\) satisfy the equation \(5 x ^ { 2 } - 60 x + 159 = 0\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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 \(B\).

9 A circle has centre at the point \(B ( 5,1 )\). The point \(A ( - 1 , - 2 )\) lies on the circle.

1. Find the equation of the circle.
   Point \(C\) is such that \(A C\) is a diameter of the circle. Point \(D\) has coordinates (5, 16).
2. Show that \(D C\) is a tangent to the circle.
   The other tangent from \(D\) to the circle touches the circle at \(E\).
3. Find the coordinates of \(E\).

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 \(A B\).

12 \includegraphics[max width=\textwidth, alt={}, center]{10b2ec29-adca-4313-ae24-bab8b2d9f8a4-18_750_981_258_580} The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 4 y - 27 = 0\) and the tangent to the circle at the point \(P ( 5,4 )\).

1. The tangent to the circle at \(P\) meets the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\). Find the area of triangle \(O A B\), where \(O\) is the origin.
2. Points \(Q\) and \(R\) also lie on the circle, such that \(P Q R\) is an equilateral triangle. Find the exact area of triangle \(P Q R\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

11 The coordinates of points \(A , B\) and \(C\) are \(A ( 5 , - 2 ) , B ( 10,3 )\) and \(C ( 2 p , p )\), where \(p\) is a constant.

1. Given that \(A C\) and \(B C\) are equal in length, find the value of the fraction \(p\).
2. It is now given instead that \(A C\) is perpendicular to \(B C\) and that \(p\) is an integer.
   1. Find the value of \(p\).
   2. Find the equation of the circle which passes through \(A , B\) and \(C\), giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\), where \(a , b\) and \(c\) are constants.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 The circle with equation \(( x - 3 ) ^ { 2 } + ( y - 5 ) ^ { 2 } = 40\) intersects the \(y\)-axis at points \(A\) and \(B\).

1. Find the \(y\)-coordinates of \(A\) and \(B\), expressing your answers in terms of surds.
2. Find the equation of the circle which has \(A B\) as its diameter.

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 \(B\).

8 \includegraphics[max width=\textwidth, alt={}, center]{71fe6352-e0dc-4c3a-8b54-99709a1782ca-3_796_695_1539_726} The diagram shows a rhombus \(A B C D\) in which the point \(A\) is ( \(- 1,2\) ), the point \(C\) is ( 5,4 ) and the point \(B\) lies on the \(y\)-axis. Find

1. the equation of the perpendicular bisector of \(A C\),
2. the coordinates of \(B\) and \(D\),
3. the area of the rhombus.

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 \(r\).

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.