Question 55
Let $a$, $b$, and $c$ be real numbers satisfying the system of equations
$$a\sqrt{1 + b^{2}} + b\sqrt{1 + a^{2}} = \frac{3}{4},$$
$$b\sqrt{1 + c^{2}} + c\sqrt{1 + b^{2}} = \frac{5}{12},$$
and
$$c\sqrt{1 + a^{2}} + a\sqrt{1 + c^{2}} = \frac{21}{20}.$$
Compute $a$.
$$a\sqrt{1 + b^{2}} + b\sqrt{1 + a^{2}} = \frac{3}{4},$$
$$b\sqrt{1 + c^{2}} + c\sqrt{1 + b^{2}} = \frac{5}{12},$$
and
$$c\sqrt{1 + a^{2}} + a\sqrt{1 + c^{2}} = \frac{21}{20}.$$
Compute $a$.