10. (20 points) Given the function
$$
f(x)=a x \sqrt{x-1}+b x+c \text {, }
$$

where, $x \in[1,+\infty), a, b, c \in \mathbf{R}$, and $a \neq 0, f(x)$ satisfies $0<2 f(5)=3 f(10)=4 f(17) \leqslant 1$. Find the maximum value of the real number $a$.
10. Let $t=\sqrt{x-1} \geqslant 0$. Then the problem is transformed into
$$
g(t)=a t^{3}+b t^{2}+a t+b+c \quad (t \in[0,+\infty))
$$

satisfying $0<2 g(2)=3 g(3)=4 g(4) \leqslant 1$, find the maximum value of the real number $a$.

Assume $g(t)=a(t-2)(t-3)(t-4)+p t^{2}+q t+r$, where $q=-25 a, b=p-9 a, b+c=r-24 a$.
Then $g(2)=4 p+2 q+r$,
$g(3)=9 p+3 q+r$,
$g(4)=16 p+4 q+r$.
By $2 g(2)=3 g(3)=4 g(4)$
$\Rightarrow q=-9 p, r=26 p$.
Thus, $g(t)=a(t-2)(t-3)(t-4)-\frac{q}{9}\left(t^{2}-9 t+26\right)$.
Given $-\frac{3}{8} \leqslant q<0$, hence $0<a \leqslant \frac{3}{200}$.
Upon verification, $f(x)=\frac{3}{200} x \sqrt{x-1}-\frac{7}{75} x-\frac{49}{60}$ satisfies the conditions.
Therefore, the maximum value of the real number $a$ is $\frac{3}{200}$.
$$
\begin{array}{l}
\leqslant \sqrt{\left(\sin ^{2 \alpha-\beta} \frac{2}{2}+\cos \frac{2 \alpha-\beta}{2}\right)\left(9 \sin \frac{2 \alpha+\beta}{2}+4 \cos ^{2} \frac{\alpha+\beta}{2}+9 \cos ^{2} \frac{\alpha+\beta}{2}+4 \sin ^{2} \frac{2+\beta}{2}\right)} \\
=\sqrt{13} \text {. } \\
\end{array}
$$

Thus, $\left(S_{\odot \mathscr{P}}\right)_{\max }=13 \pi$.