Example 5. Let $\mathrm{M}$ be the midpoint of any chord $\mathrm{AB}$ parallel to the $\mathrm{x}$-axis (not on the $\mathrm{x}$-axis) of the circle $\mathrm{x}^{2}+\mathrm{y}^{2}=1$, and let $\mathrm{C}$ be the intersection point of the known circle with the positive direction of the $x$-axis. Find the locus of the intersection point $\mathrm{P}$ of $\mathrm{OB}$ and $\mathrm{MC}$.
Analysis: As shown in Figure 5, to find the equations of the two lines OB and MC, we need to set \(\angle BOC = \theta\), obtaining the coordinates of points B and M as \((\cos \theta, \sin \theta)\) and \((0, \sin \theta)\), respectively. Point C has coordinates \((1,0)\). Therefore, the equation of line OB is \(y = x \cdot \tan(\theta + 0) (1)\); the equation of line MC is \(x + \frac{y}{\sin \theta} = 1\).

Solving equations (1) and (2) simultaneously, which actually gives the parametric equations of the trajectory of point P, and eliminating the parameter \(\theta\), we obtain the condition that the coordinates \((x, y)\) of point P must satisfy, which is \(y^2 = -2x + 1 (|y| < 1)\). This is the equation of the trajectory of point P.