This problem asks us to identify which of the given curves is always tangent to the family of lines defined by the equation $12x\cos \theta + 5y\sin \theta = 60$, for any value of $\theta$. This is a classic question involving the tangency conditions for conic sections.

Key Concepts: Tangency Conditions for Conic Sections

A line given by the equation $lx + my = n$ is tangent to:

1. A circle $x^2 + y^2 = r^2$ if and only if $r^2(l^2 + m^2) = n^2$.
2. An ellipse $\frac{x^2}{a^2} + \frac{