Followup question about product of slopes on unit circle at rational points

Question

This is a followup question to:

Product of slopes of rational points on the unit circle (related to pythagorean triples)

mathlove correctly showed that $D=1$ gives an infinity of solution pairs. But what about $D \ne 1$? Are there any values that give more than 1 solution pair? Is there some systematic way to analyze this?

Answer 1

There are multiple pairs for $D \ne 1$. Example

$D = 32/7$

$(t_1,t_2)=(1/2,3/4)$ or

$(t_1,t_2)=(4/11,5/6)$

$D = 32/45$

$(t_1,t_2)=(1/2,1/4)$ or

$(t_1,t_2)=(4/21,10/17)$

interesting.