The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\)
and the ellipse \(E\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1\)
where \(a > b > 0\)
The line \(l\) is a tangent to hyperbola \(H\) at the point \(P ( a \sec \theta , b \tan \theta )\), where \(0 < \theta < \frac { \pi } { 2 }\)
Using calculus, show that an equation for \(l\) is
$$b x \sec \theta - a y \tan \theta = a b$$
Given that the point \(F\) is the focus of ellipse \(E\) for which \(x > 0\) and that the line \(l\) passes through \(F\),