7.A circle \(C\) has centre \(X ( a , b )\) and radius \(r\) .
A line \(l\) has equation \(y = m x + c\)
- Show that the \(x\) coordinates of the points where \(C\) and \(l\) intersect satisfy
$$\left( m ^ { 2 } + 1 \right) x ^ { 2 } - 2 ( a - m ( c - b ) ) x + a ^ { 2 } + ( c - b ) ^ { 2 } - r ^ { 2 } = 0$$
Given that \(l\) is a tangent to \(C\) ,
- show that
$$c = b - m a \pm r \sqrt { m ^ { 2 } + 1 }$$
The circle \(C _ { 1 }\) has equation
$$x ^ { 2 } + y ^ { 2 } - 16 = 0$$
and the circle \(C _ { 2 }\) has equation
$$x ^ { 2 } + y ^ { 2 } - 20 x - 10 y + 89 = 0$$
- Find the equations of any lines that are normal to both \(C _ { 1 }\) and \(C _ { 2 }\) ,justifying your answer.
- Find the equations of all lines that are a tangent to both \(C _ { 1 }\) and \(C _ { 2 }\)
[You may find the following Pythagorean triple helpful in this part: \(7 ^ { 2 } + 24 ^ { 2 } = 25 ^ { 2 }\) ]