4. Let the curve $C_{1}: x^{2}+y^{2}=1$, the curve $C_{2}: x+y=1$, and $P$ be a point on $C_{2}$. The ray $O P$ intersects $C_{1}$ at $R$. Also, point $Q$ is on $O P$ and satisfies $|O Q| \cdot|O P|=|O R|^{2}$. Find the equation of the trajectory of point $Q$ as $P$ moves along the curve $C_{2}$.
4. Let $f(x, y)=x^{2}+y^{2}, g(x, y)=x+y$. Note property 9, and $f(t x, t y)=(t x)^{2}+(t y)^{2}=t^{2} f(x, y)$, $g(t x, t y)=t x+t y=t \cdot g(x, y)$, thus the equation of the trajectory of point $Q$ is $x^{2}+y^{2}=x+y(x+y>0)$, which is $\left(x-\frac{1}{2}\right)^{2}+$ $\left(y-\frac{1}{2}\right)^{2}=\frac{1}{2}$ (excluding the origin) as required.