Circle through three points

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 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 \(E\).

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.

11 The coordinates of points \(A , B\) and \(C\) are (6, 4), ( \(p , 7\) ) and (14, 18) respectively, where \(p\) is a constant. The line \(A B\) is perpendicular to the line \(B C\).

1. Given that \(p < 10\), find the value of \(p\).
   A circle passes through the points \(A , B\) and \(C\).
2. Find the equation of the circle.
3. Find the equation of the tangent to the circle at \(C\), giving the answer in the form \(d x + e y + f = 0\), where \(d , e\) and \(f\) are integers.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

10 Points \(A\) and \(B\) have coordinates \(( 4,3 )\) and \(( 8 , - 5 )\) respectively. A circle with radius 10 passes through the points \(A\) and \(B\).

1. Show that the centre of the circle lies on the line \(y = \frac { 1 } { 2 } x - 4\).
   \includegraphics[max width=\textwidth, alt={}, center]{49e137bf-42cc-41af-b5d9-85301d4699b8-14_2715_35_109_2010}
2. Find the two possible equations of the circle.

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.

6. (i) The circle \(C _ { 1 }\) has equation $$x ^ { 2 } + y ^ { 2 } + 10 x - 12 y = k \quad \text { where } k \text { is a constant }$$

1. Find the coordinates of the centre of \(C _ { 1 }\)
2. State the possible range in values for \(k\).
   (ii) The point \(P ( p , 0 )\), the point \(Q ( - 2,10 )\) and the point \(R ( 8 , - 14 )\) lie on a different circle, \(C _ { 2 }\) Given that
   • \(p\) is a positive constant
   • \(Q R\) is a diameter of \(C _ { 2 }\)
     find the exact value of \(p\).
     

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

12

1. Find the equation of the line passing through \(\mathrm { A } ( - 1,1 )\) and \(\mathrm { B } ( 3,9 )\).
2. Show that the equation of the perpendicular bisector of AB is \(2 y + x = 11\).
3. A circle has centre \(( 5,3 )\), so that its equation is \(( x - 5 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = k\). Given that the circle passes through A , show that \(k = 40\). Show that the circle also passes through B .
4. Find the \(x\)-coordinates of the points where this circle crosses the \(x\)-axis. Give your answers in surd form.

13 In Fig.13, XP and XQ are the perpendicular bisectors of AC and BC respectively. \begin{figure}[h] \end{figure}

1. Find the coordinates of X .
2. Hence show that \(\mathrm { AX } = \mathrm { BX } = \mathrm { CX }\).
3. The circumcircle of a triangle is the circle which passes through the vertices of the triangle.
   Write down the equation of the circumcircle of the triangle ABC .
4. Find the coordinates of the points where the circle cuts the \(x\) axis.

6. The points \(P , Q\) and \(R\) have coordinates (-5, 2), (-3, 8) and (9, 4) respectively.

1. Show that \(\angle P Q R = 90 ^ { \circ }\). Given that \(P , Q\) and \(R\) all lie on a circle,
2. find the coordinates of the centre of the circle,
3. show that the equation of the circle can be written in the form $$x ^ { 2 } + y ^ { 2 } - 4 x - 6 y = k$$ where \(k\) is an integer to be found.

5

1. Points A and B have coordinates \(( - 2,1 )\) and \(( 3,4 )\) respectively. Find the equation of the perpendicular bisector of AB and show that it may be written as \(5 x + 3 y = 10\).
2. Points C and D have coordinates \(( - 5,4 )\) and \(( 3,6 )\) respectively. The line through C and D has equation \(4 y = x + 21\). The point E is the intersection of CD and the perpendicular bisector of AB . Find the coordinates of point E .
3. Find the equation of the circle with centre E which passes through A and B . Show also that CD is a diameter of this circle.

10

1. Points A and B have coordinates \(( - 2,1 )\) and \(( 3,4 )\) respectively. Find the equation of the perpendicular bisector of AB and show that it may be written as \(5 x + 3 y = 10\).
2. Points C and D have coordinates \(( - 5,4 )\) and \(( 3,6 )\) respectively. The line through C and D has equation \(4 y = x + 21\). The point E is the intersection of CD and the perpendicular bisector of AB . Find the coordinates of point E .
3. Find the equation of the circle with centre E which passes through A and B . Show also that CD is a diameter of this circle.

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 \(C\).

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.

8 A circle of radius 6 passes through the points \(( 0,0 )\) and \(( 0,10 )\). 8

1. Sketch the two possible positions of the circle.
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   Show that \(\tan ^ { 2 } 15 ^ { \circ }\) can be written in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.
   Fully justify your answer.
   [0pt] [3 marks]

7

1. Find the equation of the perpendicular bisector of \(A B\)
   7
2. \(\quad\) A circle passes through the points \(A\) and \(B\) A diameter of the circle lies along the \(x\)-axis.
   Find the equation of the circle.
   [0pt] [4 marks]

9 The points \(P\) and \(Q\) have coordinates ( \(- 6,15\) ) and (12, 19) respectively. 9

1. 1. Find the coordinates of the midpoint of \(P Q\) 9
2. (ii) Find the equation of the perpendicular bisector of \(P Q\)
   Give your answer in the form \(a x + b y = c\) where \(a , b\) and \(c\) are integers.
   [0pt] [4 marks]
   9
3. 1. A circle passes through the points \(P\) and \(Q\) The centre of the circle lies on the line with equation \(2 x - 5 y = - 30\)
      Find the equation of the circle. 9
4. (ii) The circle intersects the coordinate axes at \(n\) points.
   State the value of \(n\) $$y = \sin x ^ { \circ }$$ for \(- 360 \leq x \leq 360\) is shown below.
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