Let $X$ be a norm curve in the $\mathit{SL}(2,\mathbb{C})$-character
variety of a knot exterior $M$. Let $t = \|\beta\| / \|\alpha\|$
be the ratio of the Culler-Shalen norms of two distinct non-zero
classes $\alpha, \beta \in H_1(\partial M,\mathbb{Z})$. We
demonstrate that either $X$ has exactly two associated strict
boundary slopes $\pm t$, or else there are strict boundary
slopes $r_1$ and $r_2$ with $|r_1| > t$ and $|r_2| <
t$. As a consequence, we show that there are strict boundary
slopes near cyclic, finite, and Seifert slopes. We also prove
that the diameter of the set of strict boundary slopes can
be bounded below using the Culler-Shalen norm of those slopes.