6. The hyperbola \(H\) has equation
$$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1$$
The line \(l\) is a tangent to \(H\) at the point \(P ( 4 \cosh \alpha , 2 \sinh \alpha )\), where \(\alpha\) is a constant, \(\alpha \neq 0\)
- Using calculus, show that an equation for \(l\) is
$$2 y \sinh \alpha - x \cosh \alpha + 4 = 0$$
The line \(l\) cuts the \(y\)-axis at the point \(A\).
- Find the coordinates of \(A\) in terms of \(\alpha\).
The point \(B\) has coordinates ( \(0,10 \sinh \alpha\) ) and the point \(S\) is the focus of \(H\) for which \(x > 0\)
- Show that the line segment \(A S\) is perpendicular to the line segment \(B S\).