(a) Differentiate the equation \(x^2y - ay^2 = 4a^3\) implicitly with respect to \(x\):

\(\frac{d}{dx}(x^2y) - \frac{d}{dx}(ay^2) = 0\).

Using the product rule, \(\frac{d}{dx}(x^2y) = 2xy + x^2\frac{dy}{dx}\).

For \(\frac{d}{dx}(ay^2) = 2ay\frac{dy}{dx}\).

Substitute these into the equation:

\(2xy + x^2\frac{dy}{dx} - 2ay\frac{dy}{dx} = 0\).

Rearrange to solve for \(\frac{dy}{dx}\):

\(x^2\frac{dy}{dx} - 2ay\frac{dy}{dx} = -2xy\).

\(\frac{dy}{dx}(x^2 - 2ay) = -2xy\).

\(\frac{dy}{dx} = \frac{2xy}{2ay - x^2}\).

(b) For the tangent to be parallel to the y-axis, \(\frac{dy}{dx}\) is undefined, which occurs when the denominator is zero:

\(2ay - x^2 = 0\).

\(x^2 = 2ay\).

Substitute \(x^2 = 2ay\) into the original equation:

\(2ay^2 - ay^2 = 4a^3\).

\(ay^2 = 4a^3\).

\(y^2 = 4a^2\).

\(y = 2a\) or \(y = -2a\).

For \(y = 2a\), \(x^2 = 2a(2a) = 4a^2\), so \(x = 2a\) or \(x = -2a\).

The points are \((2a, 2a)\) and \((-2a, 2a)\).