Given that \(y = x ^ { 2 } - 5 x + 15\) and \(5 x - y = 10\), show that \(x ^ { 2 } - 10 x + 25 = 0\).
Find the discriminant of \(x ^ { 2 } - 10 x + 25\).
What can you deduce from the answer to part (ii) about the line \(5 x - y = 10\) and the curve \(y = x ^ { 2 } - 5 x + 15\) ?
Solve the simultaneous equations
$$y = x ^ { 2 } - 5 x + 15 \text { and } 5 x - y = 10$$
Hence, or otherwise, find the equation of the normal to the curve \(y = x ^ { 2 } - 5 x + 15\) at the point \(( 5,15 )\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.