Rational angle bisectors on the coordinate plane and solutions of Pell's equations

Abstract

On the coordinate plane, the slopes $a$ and $b$ of two straight lines and the slope $c$ of one of their angle bisectors satisfy the equation $(a-c)^2(b^2+1) = (b-c)^2(a^2+1).$ Recently, an explicit formula for nontrivial integral solutions of this equation with solutions of negative Pell's equations was discovered by the author. In this article, for a given square-free integer $d > 1$ and a given integer $z > 1,$ we describe every integral solution $(x,y)$ of $|x^2-dy^2| = z$ such that $x$ and $dy$ are coprime by using the fundamental unit of $\mathbb Q(\sqrt d)$ and elements of $\mathbb Z[\sqrt d]$ whose absolute value of norms are the smallest prime powers. We also describe every nontrivial rational solution of the above equation as one of its applications.