5. In the Cartesian coordinate system, if the equation $m\left(x^{2}+y^{2}+2 y+1\right)=(x-2 y+3)^{2}$ represents an ellipse, then the range of values for $m$ is
A. $(0,1)$
B. $(1,+\infty)$
C. $(0.5)$
D. $(5,+\infty)$
5. D Hint: The original equation can be transformed into $\frac{\sqrt{x^{2}+(y+1)^{2}}}{\left|\frac{x-2 y+3}{\sqrt{5}}\right|}=\sqrt{\frac{5}{m}}$, which indicates that the ratio of the distance from $(x, y)$ to the fixed point $(0,-1)$ and the distance to the fixed line $x-2 y+3=0$ is a constant $\sqrt{\frac{5}{m}}$. Therefore, by the definition of an ellipse, $\sqrt{\frac{5}{m}}5$.