1. Let $x, y \in \mathbf{R}$, and the sets are
$$
\begin{array}{l}
A=\left\{(x, y) \left\lvert\, x^{2}-3 x y+4 y^{2}=\frac{7}{2}\right.\right\}, \\
B=\{(x, y) \mid k x+y=2\}(k>0) .
\end{array}
$$

When $A \cap B$ has only one element, $k=$ $\qquad$
-1. $\frac{1}{4}$.
From $\left\{\begin{array}{l}x^{2}-3 x y+4 y^{2}=\frac{7}{2} \\ k x+y=2\end{array}\right.$,
$$
\Rightarrow\left(4 k^{2}+3 k+1\right) x^{2}-2(3+8 k) x+\frac{25}{2}=0 \text {. }
$$

Since $A \cap B$ has only one element, we have,
$$
\begin{array}{l}
\Delta=4(3+8 k)^{2}-50\left(4 k^{2}+3 k+1\right)=0 \\
\Rightarrow 4 k^{2}+3 k-1=0 \\
\Rightarrow(4 k-1)(k+1)=0 \\
\Rightarrow k=\frac{1}{4} .
\end{array}
$$