Normal to circle at point

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } + 8 x - 4 y = 0$$

1. Find
   1. the coordinates of the centre of \(C\),
   2. the exact radius of \(C\). The point \(P\) lies on \(C\).
      Given that the tangent to \(C\) at \(P\) has equation \(x + 2 y + 10 = 0\)
2. find the coordinates of \(P\)
3. Find the equation of the normal to \(C\) at \(P\), giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are integers to be found.

6 The diagram shows a curve and a line which intersect at the points \(A , B\) and \(C\).
\includegraphics[max width=\textwidth, alt={}, center]{f2124c89-79de-4758-b7b8-ff273345b9dd-7_574_844_349_609} The curve has equation \(y = x ^ { 3 } - x ^ { 2 } - 5 x + 7\) and the straight line has equation \(y = x + 7\). The point \(B\) has coordinates ( 0,7 ).

1. 1. Show that the \(x\)-coordinates of the points \(A\) and \(C\) satisfy the equation $$x ^ { 2 } - x - 6 = 0$$
   2. Find the coordinates of the points \(A\) and \(C\).
2. Find \(\int \left( x ^ { 3 } - x ^ { 2 } - 5 x + 7 \right) \mathrm { d } x\).
3. Find the area of the shaded region \(R\) bounded by the curve and the line segment \(A B\).
   [0pt] [4 marks]
   \(7 \quad\) A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 10 x + 12 y + 41 = 0\). The point \(A ( 3 , - 2 )\) lies on the circle.

11

1. Show that the equation of the radius of the circle through \(P\) is \(y = 7 x - 37\)
   

   11	Find the equation of \(C\)		Do not write outside the box
   		[4 marks]