15. Let $\boldsymbol{i}, \boldsymbol{j}$ be the unit vectors in the positive directions of the $x, y$ axes in a Cartesian coordinate system. If the vectors $\boldsymbol{a}=(x+2) \boldsymbol{i}+y \boldsymbol{j}$, $\boldsymbol{b}=(x-2) \boldsymbol{i}+y \boldsymbol{j}$, and $|\boldsymbol{a}|-|\boldsymbol{b}|=2$.
(1) Find the equation of the locus of point $P(x, y)$ that satisfies the above conditions;
(2) Let $A(-1,0), F(2,0)$. Does there exist a constant $\lambda(\lambda>0)$ such that $\angle P F A=\lambda \angle P A F$ always holds? Prove your conclusion.
(1) According to the definition of a hyperbola, we have \(a=1, c=2, b=\sqrt{3}\). The trajectory equation of point \(P(x, y)\) is \(x^{2}-\frac{y^{2}}{3}=1(x>0)\).

(2) Let \(P(\sec \theta, \sqrt{3} \tan \theta)\), then \(\tan \angle P A F=\frac{\sqrt{3} \tan \theta}{\sec \theta+1}\)
\[
\begin{array}{l}
=\frac{\sqrt{3} \sin \theta}{\cos \theta+1}, \tan \angle P F A=\frac{\sqrt{3} \tan \theta}{2-\sec \theta}=\frac{\sqrt{3} \sin \theta}{2 \cos \theta-1}. \\
\text { Since } \tan 2 \angle P A F=\frac{\frac{2 \sqrt{3} \sin \theta}{\cos \theta+1}}{1-\left(\frac{\sqrt{3} \sin \theta}{\cos \theta+1}\right)^{2}} \\
=\frac{2 \sqrt{3} \sin \theta(\cos \theta+1)}{(\cos \theta+1)^{2}-3 \sin ^{2} \theta}=\frac{\sqrt{3} \sin \theta(\cos \theta+1)}{2 \cos ^{2} \theta+\cos \theta-1}=\frac{\sqrt{3} \sin \theta}{2 \cos \theta-1}=\tan \angle P F A,
\end{array}
\]

Therefore, there exists a constant \(\lambda=2\), such that \(\angle P F A=2 \angle P A F\) always holds.