- The ellipse \(E\) has equation
$$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$
The line \(l _ { 1 }\) is a tangent to \(E\) at the point \(P ( a \cos \theta , b \sin \theta )\).
- Using calculus, show that an equation for \(l _ { 1 }\) is
$$\frac { x \cos \theta } { a } + \frac { y \sin \theta } { b } = 1$$
The circle \(C\) has equation
$$x ^ { 2 } + y ^ { 2 } = a ^ { 2 }$$
The line \(l _ { 2 }\) is a tangent to \(C\) at the point \(Q ( a \cos \theta , a \sin \theta )\).
- Find an equation for the line \(l _ { 2 }\).
Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(R\),
- find, in terms of \(a , b\) and \(\theta\), the coordinates of \(R\).
- Find the locus of \(R\), as \(\theta\) varies.