6. The rectangular hyperbola \(H\) has cartesian equation \(x y = c ^ { 2 }\).
The point \(P \left( c t , \frac { c } { t } \right) , t > 0\), is a general point on \(H\).
- Show that an equation of the tangent to \(H\) at the point \(P\) is
$$t ^ { 2 } y + x = 2 c t$$
An equation of the normal to \(H\) at the point \(P\) is \(t ^ { 3 } x - t y = c t ^ { 4 } - c\)
Given that the normal to \(H\) at \(P\) meets the \(x\)-axis at the point \(A\) and the tangent to \(H\) at \(P\) meets the \(x\)-axis at the point \(B\),
- find, in terms of \(c\) and \(t\), the coordinates of \(A\) and the coordinates of \(B\).
Given that \(c = 4\),
- find, in terms of \(t\), the area of the triangle \(A P B\). Give your answer in its simplest form.