Example 4. A tangent line is drawn through the point $\mathrm{P}(1,-1)$ to the parabola $\mathrm{y}^{2}-2 \mathrm{x}$ $-2 y+3=0$. Find the equation of the tangent line.
Correct answer: When the slope exists, the tangent line is $x+4 y+3=0$, and when the slope does not exist, the tangent line is $x=1$.

From this example, we can see that when using the point-slope form, it is necessary to consider the condition for the point-slope equation to hold, which is that the slope must exist. If the problem requires finding the equation of a line passing through a fixed point, then while using the point-slope form, one must also consider the case where the slope does not exist, i.e., the line passing through the fixed point and perpendicular to the $\mathrm{x}$-axis. Similarly, when using the slope-intercept form, it is also necessary to consider the case where the slope does not exist, i.e., the equation of the line perpendicular to the $\mathbf{x}$-axis, to avoid missing solutions; when using the two-point form $\frac{y-y_{1}}{y_{1}-y_{2}}=\frac{x-x_{1}}{x_{1}-x_{2}}$, or the intercept form $\frac{x}{a}+\frac{y}{b}=1$, there are specific conditions, i.e., the denominator cannot be 0.