Do either $(1)$ or $(2)$:

$(1)$   Let $C_a$ be the curve $(y - a^2)^2 = x^2(a^2 - x^2).$ Find the curve which touches all $C_a$ for $a > 0.$ Sketch the solution and at least two of the $C_a.$

$(2)$   Given that $(1 - hx)^{-1}(1 - kx)^{-1} = \sum_{i\geq0}a_i x^i,$ prove that $(1 + hkx)(1 - hkx)^{-1}(1 - h^2x)^{-1}(1 - k^2x)^{-1} = \sum_{i\geq0}  a_i^2 x^i.$