$25 \cdot 10$ If $(a, b)$ and $(c, d)$ are points on the line $y=m x+k$, then the distance between $(a, b)$ and $(c, d)$ expressed in terms of $a, c, m$ is
(A) $|a-c| \sqrt{1+m^{2}}$.
(B) $|a+c| \sqrt{1+m^{2}}$.
(C) $\frac{|a-c|}{\sqrt{1+m^{2}}}$.
(D) $|a-c|\left(1+m^{2}\right)$.
(E) $|a-c||m|$.
(25th American High School Mathematics Examination, 1974)
[ Sol] Since $(a, b)$ and $(c, d)$ are on the same line $y=m x+k$, they satisfy the same equation, so $b=m a+k, \quad d=m c+k$.
The distance between points $(a, b)$ and $(c, d)$ is $\sqrt{(a-c)^{2}+(b-d)^{2}}$.
From the above two equations, we get $(b-d)=m(a-c)$, so
$$
\begin{aligned}
\sqrt{(a-c)^{2}+(b-d)^{2}} & =\sqrt{(a-c)^{2}+m^{2}(a-c)^{2}} \\
& =|a-c| \sqrt{1+m^{2}} .
\end{aligned}
$$

Therefore, the answer is $(A)$.