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

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 \(B\).

7 The point \(A\) has coordinates \(( 1,5 )\) and the line \(l\) has gradient \(- \frac { 2 } { 3 }\) and passes through \(A\). A circle has centre \(( 5,11 )\) and radius \(\sqrt { 52 }\).

1. Show that \(l\) is the tangent to the circle at \(A\).
2. Find the equation of the other circle of radius \(\sqrt { 52 }\) for which \(l\) is also the tangent at \(A\).

10 Points \(A ( - 2,3 ) , B ( 3,0 )\) and \(C ( 6,5 )\) lie on the circumference of a circle with centre \(D\).

1. Show that angle \(A B C = 90 ^ { \circ }\).
2. Hence state the coordinates of \(D\).
3. Find an equation of the circle.
   The point \(E\) lies on the circumference of the circle such that \(B E\) is a diameter.
4. Find an equation of the tangent to the circle at \(E\).

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.

7 \includegraphics[max width=\textwidth, alt={}, center]{89a18f20-a4d6-4a42-8b00-849f4fb89692-10_887_1003_258_571} The diagram shows the circle with equation \(( x - 2 ) ^ { 2 } + ( y + 4 ) ^ { 2 } = 20\) and with centre \(C\). The point \(B\) has coordinates \(( 0,2 )\) and the line segment \(B C\) intersects the circle at \(P\).

1. Find the equation of \(B C\).
2. Hence find the coordinates of \(P\), giving your answer in exact form.

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 \(B\).

5

1. Express \(2 x ^ { 2 } - 8 x + 14\) in the form \(2 \left[ ( x - a ) ^ { 2 } + b \right]\).
   The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } ( x ) = x ^ { 2 } \quad \text { for } x \in \mathbb { R } \\ & \mathrm {~g} ( x ) = 2 x ^ { 2 } - 8 x + 14 \quad \text { for } x \in \mathbb { R } \end{aligned}$$
2. Describe fully a sequence of transformations that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { g } ( x )\), making clear the order in which the transformations are applied. \includegraphics[max width=\textwidth, alt={}, center]{05e75fa2-81ae-44b1-b073-4100f5d911e0-08_679_1043_260_552} The circle with equation \(( x + 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 85\) and the straight line with equation \(y = 3 x - 20\) are shown in the diagram. The line intersects the circle at \(A\) and \(B\), and the centre of the circle is at \(C\).

8 \includegraphics[max width=\textwidth, alt={}, center]{05e75fa2-81ae-44b1-b073-4100f5d911e0-12_771_839_262_651} The diagram shows the circle with equation \(( x - 2 ) ^ { 2 } + y ^ { 2 } = 8\). The chord \(A B\) of the circle intersects the positive \(y\)-axis at \(A\) and is parallel to the \(x\)-axis.

1. Find, by calculation, the coordinates of \(A\) and \(B\).
2. Find the volume of revolution when the shaded segment, bounded by the circle and the chord \(A B\), is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.

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 \(B\).

9 \includegraphics[max width=\textwidth, alt={}, center]{fdd6e942-b5bc-4369-8587-6de120459776-12_583_661_260_740} The diagram shows a circle with centre \(A\) passing through the point \(B\). A second circle has centre \(B\) and passes through \(A\). The tangent at \(B\) to the first circle intersects the second circle at \(C\) and \(D\). The coordinates of \(A\) are ( \(- 1,4\) ) and the coordinates of \(B\) are ( 3,2 ).

1. Find the equation of the tangent CBD.
2. Find an equation of the circle with centre \(B\).
3. Find, by calculation, the \(x\)-coordinates of \(C\) and \(D\). \includegraphics[max width=\textwidth, alt={}, center]{fdd6e942-b5bc-4369-8587-6de120459776-14_746_652_262_744} The diagram shows a sector \(C A B\) which is part of a circle with centre \(C\). A circle with centre \(O\) and radius \(r\) lies within the sector and touches it at \(D , E\) and \(F\), where \(C O D\) is a straight line and angle \(A C D\) is \(\theta\) radians.

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 \(E\).

11 A circle with centre \(C\) has equation \(( x - 8 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 100\).

1. Show that the point \(T ( - 6,6 )\) is outside the circle.
   Two tangents from \(T\) to the circle are drawn.
2. Show that the angle between one of the tangents and \(C T\) is exactly \(45 ^ { \circ }\).
   The two tangents touch the circle at \(A\) and \(B\).
3. Find the equation of the line \(A B\), giving your answer in the form \(y = m x + c\).
4. Find the \(x\)-coordinates of \(A\) and \(B\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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.

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 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 \(B\).

8 \includegraphics[max width=\textwidth, alt={}, center]{8eb3d21b-dc45-493c-9e5c-3c0535c505e8-10_492_888_255_625} The diagram shows two identical circles intersecting at points \(A\) and \(B\) and with centres at \(P\) and \(Q\). The radius of each circle is \(r\) and the distance \(P Q\) is \(\frac { 5 } { 3 } r\).

1. Find the perimeter of the shaded region in terms of \(r\).
2. Find the area of the shaded region in terms of \(r\).

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.