8. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1\).
The line \(l _ { 1 }\) is the tangent to \(H\) at the point \(P ( 4 \sec t , 2 \tan t )\).
- Use calculus to show that an equation of \(l _ { 1 }\) is
$$2 y \sin t = x - 4 \cos t$$
The line \(l _ { 2 }\) passes through the origin and is perpendicular to \(l _ { 1 }\).
The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(Q\). - Show that, as \(t\) varies, an equation of the locus of \(Q\) is
$$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 16 x ^ { 2 } - 4 y ^ { 2 }$$