Parte (a):
1. Derivamos implícitamente $\sqrt{x} + \sqrt{y} = \sqrt{a}$:
$$ \frac{1}{2\sqrt{x}} + \frac{1}{2\sqrt{y}} \frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{\sqrt{y}}{\sqrt{x}} $$
2. En un punto $(x_0, y_0)$, la tangente es: $y - y_0 = -\frac{\sqrt{y_0}}{\sqrt{x_0}}(x - x_0)$.
3. Intercepto $x$ ($y=0$): $-y_0 = -\frac{\sqrt{y_0}}{\sqrt{x_0}}x + \sqrt{x_0 y_0} \implies x_{int} = \sqrt{x_0 y_0} \frac{\sqrt{x_0}}{\sqrt{y_0}} + x_0 = x_0 + \sqrt{x_0 y_0} = \sqrt{x_0}(\sqrt{x_0} + \sqrt{y_0})$.
Como $\sqrt{x_0} + \sqrt{y_0} = \sqrt{a}$, entonces $x_{int} = \sqrt{ax_0}$.
4. Intercepto $y$ ($x=0$): $y_{int} = y_0 + \sqrt{x_0 y_0} = \sqrt{y_0}(\sqrt{y_0} + \sqrt{x_0}) = \sqrt{ay_0}$.
5. Suma: $S = \sqrt{ax_0} + \sqrt{ay_0} = \sqrt{a}(\sqrt{x_0} + \sqrt{y_0}) = \sqrt{a}(\sqrt{a}) = a$.

$$ \boxed{S = a \text{ (Constante)}} $$

Parte (b):
1. Derivamos $x^{2/3} + y^{2/3} = a^{2/3}$: $\frac{2}{3}x^{-1/3} + \frac{2}{3}y^{-1/3}y' = 0 \implies y' = -\left(\frac{y}{x}\right)^{1/3}$.
2. Ecuación tangente en $(x_0, y_0)$: $y - y_0 = -\frac{y_0^{1/3}}{x_0^{1/3}}(x - x_0)$.
3. Intercepto $x$ ($y=0$): $x_{int} = x_0 + y_0 \frac{x_0^{1/3}}{y_0^{1/3}} = x_0^{1/3}(x_0^{2/3} + y_0^{2/3}) = x_0^{1/3} a^{2/3}$.
4. Intercepto $y$ ($x=0$): $y_{int} = y_0 + x_0 \frac{y_0^{1/3}}{x_0^{1/3}} = y_0^{1/3}(y_0^{2/3} + x_0^{2/3}) = y_0^{1/3} a^{2/3}$.
5. Suma de cuadrados: $L^2 = (x_0^{1/3} a^{2/3})^2 + (y_0^{1/3} a^{2/3})^2 = a^{4/3}(x_0^{2/3} + y_0^{2/3}) = a^{4/3} a^{2/3} = a^2$.

$$ \boxed{L^2 = a^2 \text{ (Constante)}} $$