The line \(l _ { 1 }\) passes through the point \(A ( 2,5 )\) and has gradient \(- \frac { 1 } { 2 }\).
Find an equation of \(l _ { 1 }\), giving your answer in the form \(y = m x + c\).
The point \(B\) has coordinates (-2, 7).
Show that \(B\) lies on \(l _ { 1 }\).
Find the length of \(A B\), giving your answer in the form \(k \sqrt { } 5\), where \(k\) is an integer.
The point \(C\) lies on \(l _ { 1 }\) and has \(x\)-coordinate equal to \(p\).
The length of \(A C\) is 5 units.
Show that \(p\) satisfies
$$p ^ { 2 } - 4 p - 16 = 0 .$$