Distance from centre to line

8. The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 2 x - 18 y + 73 = 0\) and the straight line with equation \(y = 2 x - 3\).

1. Find the coordinates of the centre and the radius of the circle.
2. Find the coordinates of the point on the line which is closest to the circle.

11. \begin{figure}[h] \end{figure} Figure 3 shows the circle \(C\) with equation $$x ^ { 2 } + y ^ { 2 } - 10 x - 8 y + 32 = 0$$ and the line \(l\) with equation $$2 y + x + 6 = 0$$

1. Find
   1. the coordinates of the centre of \(C\),
   2. the radius of \(C\).
2. Find the shortest distance between \(C\) and \(l\).

6 In this question you must show detailed reasoning. The diagram shows the line \(3 y + x = 7\) which is a tangent to a circle with centre \(( 3 , - 2 )\).
Find an equation for the circle.

7 A line has equation \(y = 2 x\) and a circle has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 16 y + 56 = 0\).

1. Show that the line does not meet the circle.
2. (a) Find the equation of the line through the centre of the circle that is perpendicular to the line \(y = 2 x\).
   (b) Hence find the shortest distance between the line \(y = 2 x\) and the circle, giving your answer in an exact form.

8 The lines \(L _ { 1 }\) and \(L _ { 2 }\) are parallel. \(L _ { 1 }\) has equation $$5 x + 3 y = 15$$ and \(L _ { 2 }\) has equation $$5 x + 3 y = 83$$ \(L _ { 1 }\) intersects the \(y\)-axis at the point \(P\).
The point \(Q\) is the point on \(L _ { 2 }\) closest to \(P\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-10_849_917_945_561} 8

1. 1. Find the coordinates of \(Q\).
      8
      1. (ii) Hence show that \(P Q = k \sqrt { 34 }\), where \(k\) is an integer to be found. 8
   2. A circle, \(C\), has centre ( \(a , - 17\) ). \(L _ { 1 }\) and \(L _ { 2 }\) are both tangents to \(C\).
      8
   3. 1. Find \(a\).
         8
   4. (ii) Find the equation of \(C\). \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-13_2493_1732_214_139}

In this question you must show detailed reasoning. The centre of a circle C is at the point \((-1, 3)\) and C passes through the point \((1, -1)\). The straight line L passes through the points \((1, 9)\) and \((4, 3)\). Show that L is a tangent to C. [7]

In this question you must show detailed reasoning. The centre of a circle C is the point (-1, 3) and C passes through the point (1, -1). The straight line L passes through the points (1, 9) and (4, 3). Show that L is a tangent to C. [7]

A line has equation \(y = 2x\) and a circle has equation \(x^2 + y^2 + 2x - 16y + 56 = 0\).

1. Show that the line does not meet the circle. [3]
2. 1. Find the equation of the line through the centre of the circle that is perpendicular to the line \(y = 2x\). [4]
   2. Hence find the shortest distance between the line \(y = 2x\) and the circle, giving your answer in an exact form. [4]