3. Let the line $l$ passing through $A(2,1)$ intersect the hyperbola $x^{2}-\frac{y^{2}}{2}=1$ at points $P_{1}, P_{2}$. Find the equation of the locus of the midpoint of segment $P_{1} P_{2}$.
3. Let point \(P\left(x_{0}, y_{0}\right)\) be the midpoint of line segment \(P_{1} P_{2}\), then the curve equation symmetric to \(x^{2}-\frac{y^{2}}{2}=1\) about point \(P\) is \(\left(2 x_{0}-x\right)^{2}- \frac{1}{2}\left(2 y_{0}-y\right)^{2}=1\). Subtracting the two equations yields \(2 x_{0} x-y_{0} y-2 x_{0}^{2}+y_{0}^{2}=0\). Since point \(A(2,1)\) lies on this line, we have \(2 x_{0}^{2}- y_{0}^{2}-4 x_{0}+y_{0}=0\), thus the equation of the locus of the midpoint is \(2 x^{2}-y^{2}-4 x+y=0\).