7 A circle has equation \(x ^ { 2 } + y ^ { 2 } - 4 x - 14 = 0\).
- Find:
- the coordinates of the centre of the circle;
- the radius of the circle in the form \(p \sqrt { 2 }\), where \(p\) is an integer.
- A chord of the circle has length 8. Find the perpendicular distance from the centre of the circle to this chord.
- A line has equation \(y = 2 k - x\), where \(k\) is a constant.
- Show that the \(x\)-coordinate of any point of intersection of the line and the circle satisfies the equation
$$x ^ { 2 } - 2 ( k + 1 ) x + 2 k ^ { 2 } - 7 = 0$$
- Find the values of \(k\) for which the equation
$$x ^ { 2 } - 2 ( k + 1 ) x + 2 k ^ { 2 } - 7 = 0$$
has equal roots.
- Describe the geometrical relationship between the line and the circle when \(k\) takes either of the values found in part (c)(ii).