  Let $a$ and $b=ka$ be positive integers with $k\in \{2, 3, 6\},$ such that
$ab+4$ is a perfect square. In this paper, we study the extensibility of the
$D(4)$-pairs $\{a, ka\}.$ More precisely, we prove that by considering three
families of positive integers $c$ depending on $a,$ if $\{a, b, c, d\}$ is the
set of positive integers which has the property that the product of any two of
its elements increased by $4$ is a perfect square, then $d$ in given by
$$d=a+b+c+\frac{1}{2}\left(abc\pm \sqrt{(ab+4)(ac+4)(bc+4)}\right).$$ As a
corollary, we prove that any $D(4)$-quadruple which contains the pair $\{a,
ka\}$ is regular.
