13. The curve \(C\) has equation
$$y = 3 x ^ { 2 } - 4 x + 2$$
The line \(l _ { 1 }\) is the normal to the curve \(C\) at the point \(P ( 1,1 )\)
- Show that \(l _ { 1 }\) has equation
$$x + 2 y - 3 = 0$$
The line \(l _ { 1 }\) meets curve \(C\) again at the point \(Q\).
- By solving simultaneous equations, determine the coordinates of the point \(Q\).
Another line \(l _ { 2 }\) has equation \(k x + 2 y - 3 = 0\), where \(k\) is a constant.
- Show that the line \(l _ { 2 }\) meets the curve \(C\) once only when
$$k ^ { 2 } - 16 k + 40 = 0$$
- Find the two exact values of \(k\) for which \(l _ { 2 }\) is a tangent to \(C\).