Circle through three points using perpendicular bisectors

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.

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 \(C\).

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
      1. (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
   2. 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
   3. (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. \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-12_613_1552_532_246}

\includegraphics{figure_1} Fig. 11 shows the line through the points A \((-1, 3)\) and B \((5, 1)\).

1. Find the equation of the line through A and B. [3]
2. Show that the area of the triangle bounded by the axes and the line through A and B is \(\frac{32}{3}\) square units. [2]
3. Show that the equation of the perpendicular bisector of AB is \(y = 3x - 4\). [3]
4. A circle passing through A and B has its centre on the line \(x = 3\). Find the centre of the circle and hence find the radius and equation of the circle. [4]

The line \(L_1\) has equation \(x + 7y - 41 = 0\) \(L_1\) is a tangent to the circle C at the point P(6, 5) The line \(L_2\) has equation \(y = x + 3\) \(L_2\) is a tangent to the circle C at the point Q(0, 3) The lines \(L_1\) and \(L_2\) and the circle C are shown in the diagram below. \includegraphics{figure_11}

1. Show that the equation of the radius of the circle through P is \(y = 7x - 37\) [3 marks]
2. Find the equation of C [4 marks]

The point \(A\) has coordinates \((-2, 5)\) and the point \(B\) has coordinates \((3, 8)\). The point \(C\) lies on the \(x\)-axis such that \(AC\) is perpendicular to \(AB\).

1. Find the equation of \(AB\). [3]
2. Show that \(C\) has coordinates \((1, 0)\). [3]
3. Calculate the area of triangle \(ABC\). [4]
4. Find the equation of the circle which passes through the points \(A\), \(B\) and \(C\). [5]