I'm afraid the following is rather localized questions, but I don't know how to simplify and generalize it. I have the following two points (in the real plane):
p = {Sqrt[b y1^2 + 4 ϵ], y1}
q = {(3 b y1^2 + 12 ϵ + Sqrt[5] y1 Sqrt[b (b y1^2 + 4 ϵ)])/(2 Sqrt[b y1^2 + 4 ϵ]),
(3 b y1 + Sqrt[5] Sqrt[b (b y1^2 + 4 ϵ)])/(2 b)}
In particular, I have also the following assumptions:
$Assumptions = b > 0 && y1 ∈ Reals && ϵ > 0;
Then I try to solve the following:
Solve[p[[2]] == -q[[2]] && p[[1]] == q[[1]], y1]
and obtain the answer
{{y1 -> -(Sqrt[ϵ]/Sqrt[b])}, {y1 -> Sqrt[ϵ]/Sqrt[b]}}
However, one can easily test that the second answer is wrong:
In: {p, q} /. % // Simplify
returns: $\left\{\left\{\left\{\sqrt{5} \sqrt{\epsilon },-\sqrt{\frac{\epsilon }{b}}\right\},\left\{\sqrt{5} \sqrt{\epsilon },\sqrt{\frac{\epsilon }{b}}\right\}\right\},\left\{\left\{\sqrt{5} \sqrt{\epsilon },\sqrt{\frac{\epsilon }{b}}\right\},\left\{2 \sqrt{5} \sqrt{\epsilon },\frac{4 \epsilon }{\sqrt{b \epsilon }}\right\}\right\}\right\}$
What am I missing?