Example 3. The line $\left\{\begin{array}{c}x=2+\frac{1}{2} t \\ y=1+\frac{\sqrt{3}}{2} t\end{array}\right.$ intersects the parabola $y^{2}=4 x$ to form a chord. Find the length of the chord.
Solve $\left\{\begin{array}{l}x=2+\frac{1}{2} t, \\ y=1+\frac{\sqrt{3}}{2} t\end{array}\right.$
with the equation $y^{2}=4 x$ to get
$$
\begin{aligned}
\frac{3}{4} t^{2}-(2 & -\sqrt{3}) t-7=0 \\
\left|t_{2}-t_{1}\right| & =\sqrt{\left(t_{1}+t_{2}\right)^{2}-4 t_{1} t_{2}} \\
& =\frac{8}{3} \sqrt{7-\sqrt{3}} .
\end{aligned}
$$

Since in this problem $a^{2}+b^{2}=1$,
the chord length $\mathrm{d}=\frac{8}{3} \sqrt{7-\sqrt{3}}$.