Tangent equation at a point

2 Points \(A\) and \(B\) have coordinates \(( 3,0 )\) and \(( 9,8 )\) respectively. The line \(A B\) is a diameter of a circle.

1. Find the coordinates of the centre of the circle.
2. Find the equation of the tangent to the circle at the point \(B\).

19. The circle \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 16 y - 209 = 0\).

1. Find the coordinates of the centre of \(C\) and the radius of \(C\). The point \(P ( x , y )\) lies on \(C\).
2. Find, in terms of \(x\) and \(y\), the gradient of the tangent to \(C\) at \(P\).
3. Hence or otherwise, find an equation of the tangent to \(C\) at the point (21, 8).

5 A circle with centre \(C\) has equation \(( x + 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 25\).

1. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle.
2. 1. Verify that the point \(N ( 0 , - 2 )\) lies on the circle.
   2. Sketch the circle.
   3. Find an equation of the normal to the circle at the point \(N\).
3. The point \(P\) has coordinates (2, 6).
   1. Find the distance \(P C\), leaving your answer in surd form.
   2. Find the length of a tangent drawn from \(P\) to the circle.

5 A circle with centre \(C\) has equation $$( x - 5 ) ^ { 2 } + ( y + 12 ) ^ { 2 } = 169$$

1. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle.
2. 1. Verify that the circle passes through the origin \(O\).
   2. Given that the circle also passes through the points \(( 10,0 )\) and \(( 0 , p )\), sketch the circle and find the value of \(p\).
3. The point \(A ( - 7 , - 7 )\) lies on the circle.
   1. Find the gradient of \(A C\).
   2. Hence find an equation of the tangent to the circle at the point \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

2 A circle has equation \(( x - 2 ) ^ { 2 } + ( y + 3 ) ^ { 2 } = 13\)
Find the gradient of the tangent to this circle at the origin.
Circle your answer.
[0pt] [1 mark]
\(- \frac { 3 } { 2 }\)
\(- \frac { 2 } { 3 }\)
\(\frac { 2 } { 3 }\)
\(\frac { 3 } { 2 }\)

1 Line \(L\) has equation $$5 y = 4 x + 6$$ Find the gradient of a line parallel to line \(L\)
Circle your answer.
\(- \frac { 5 } { 4 }\)
\(- \frac { 4 } { 5 }\)
\(\frac { 4 } { 5 }\)
\(\frac { 5 } { 4 }\)

11 The circle with equation \(( x - 7 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 5\) has centre \(C\). 11

1. 1. Write down the radius of the circle. 11
2. (ii) Write down the coordinates of \(C\).
   [0pt] [1 mark] 11
3. The point \(P ( 5 , - 1 )\) lies on the circle.
   Find the equation of the tangent to the circle at \(P\), giving your answer in the form \(y = m x + c\)
   [0pt] [4 marks] 11
4. The point \(Q ( 3,3 )\) lies outside the circle and the point \(T\) lies on the circle such that \(Q T\) is a tangent to the circle. Find the length of \(Q T\).
   [0pt] [4 marks]

1 Find the gradient of the line with equation \(2 x + 5 y = 7\)
Circle your answer.
[0pt] [1 mark] $$\begin{array} { l l l l } \frac { 2 } { 5 } & \frac { 5 } { 2 } & - \frac { 2 } { 5 } & - \frac { 5 } { 2 } \end{array}$$