12 For any positive parameter \(k\), the curve \(C _ { k }\) is defined by the polar equation
\(\mathrm { r } = \mathrm { k } ( \cos \theta + 1 ) + \frac { 10 } { \mathrm { k } } , 0 \leqslant \theta \leqslant 2 \pi\).
For each value of \(k\) the curve is a single, closed loop with no self-intersections. The diagram shows \(C _ { 10.5 }\) for the purpose of illustration.
\includegraphics[max width=\textwidth, alt={}, center]{fbb82fa2-b316-44ae-a19e-197b45f51c87-6_558_723_550_242} Each curve, \(C _ { k }\), encloses a certain area, \(A _ { k }\).
You are given that there is a single minimum value of \(A _ { k }\).
Determine, in an exact form, the value of \(k\) for which \(C _ { k }\) encloses this minimum area.