Example 15 (1997 National High School League Question) In the Cartesian coordinate system, if the equation $m\left(x^{2}+y^{2}+2 y+\right.$ $1)=(x-2 y+3)^{2}$ represents an ellipse, then the range of values for $m$ is $(\quad)$.
A. $(0,1)$
B. $(1,+\infty)$
C. $(0,5)$
D. $(5,+\infty)$
Solution: Choose D. Reason: From the given, we have $\frac{\sqrt{x^{2}+(y+1)^{2}}}{\left|\frac{x-2 y+3}{\sqrt{1^{2}+(-2)^{2}}}\right|}=\sqrt{\frac{5}{m}}$, which indicates that the ratio of the distance from the moving point $(x, y)$ to the fixed point $(0, -1)$ and the distance to the line $x-2 y+3=0$ is a constant $\sqrt{\frac{5}{m}}$. By the definition of an ellipse, $\sqrt{\frac{5}{m}}5$, hence the choice is D.