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 $(1 + hkx)(1 - hkx)^{-1}(1 - h^2x)^{-1}(1 - k^2x)^{-1} = \sum_{i\geq0}  a_i^2 x^i.$
We are given the curve \( C_a \) defined by the equation:
\[
(y - a^2)^2 = x^2(a^2 - x^2)
\]

We need to find the curve which touches all \( C_a \) for \( a > 0 \).

1. **Differentiate the given equation with respect to \( a \):**

   \[
   \frac{d}{da} \left( (y - a^2)^2 \right) = \frac{d}{da} \left( x^2(a^2 - x^2) \right)
   \]

   Using the chain rule on the left-hand side:
   \[
   2(y - a^2) \cdot \frac{d}{da}(y - a^2) = 2(y - a^2) \cdot (-2a)
   \]

   On the right-hand side, we use the product rule:
   \[
   \frac{d}{da} \left( x^2(a^2 - x^2) \right) = x^2 \cdot \frac{d}{da}(a^2 - x^2) = x^2 \cdot 2a
   \]

   Equating both sides:
   \[
   2(y - a^2) \cdot (-2a) = x^2 \cdot 2a
   \]

2. **Simplify the equation:**

   \[
   -4a(y - a^2) = 2ax^2
   \]

   Divide both sides by \( 2a \) (assuming \( a \neq 0 \)):
   \[
   -2(y - a^2) = x^2
   \]

   \[
   -2y + 2a^2 = x^2
   \]

   \[
   2a^2 = x^2 + 2y
   \]

   \[
   a^2 = \frac{x^2}{2} + y
   \]

3. **Substitute \( a^2 \) back into the original equation:**

   \[
   (y - (\frac{x^2}{2} + y))^2 = x^2((\frac{x^2}{2} + y) - x^2)
   \]

   Simplify the left-hand side:
   \[
   (y - \frac{x^2}{2} - y)^2 = (-\frac{x^2}{2})^2
   \]

   \[
   \left( -\frac{x^2}{2} \right)^2 = \frac{x^4}{4}
   \]

   Simplify the right-hand side:
   \[
   x^2 \left( \frac{x^2}{2} + y - x^2 \right) = x^2 \left( y - \frac{x^2}{2} \right)
   \]

   Equate both sides:
   \[
   \frac{x^4}{4} = x^2 \left( y - \frac{x^2}{2} \right)
   \]

   Divide both sides by \( x^2 \) (assuming \( x \neq 0 \)):
   \[
   \frac{x^2}{4} = y - \frac{x^2}{2}
   \]

   \[
   y = \frac{x^2}{4} + \frac{x^2}{2}
   \]

   \[
   y = \frac{3x^2}{4}
   \]

Thus, the curve which touches all \( C_a \) for \( a > 0 \) is:
\[
y = \frac{3x^2}{4}
\]

The final answer is \(\boxed{y = \frac{3x^2}{4}}\).