Find the real values of $p$ for which the equation \[\sqrt{2p+ 1 - x^2} +\sqrt{3x + p + 4} = \sqrt{x^2 + 9x+ 3p + 9}\] in $x$ has exactly two real distinct roots.($\sqrt t $ means the positive square root of $t$).
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Tags: parametric equation, IMO Shortlist, IMO 1979
Find the real values of $p$ for which the equation \[\sqrt{2p+ 1 - x^2} +\sqrt{3x + p + 4} = \sqrt{x^2 + 9x+ 3p + 9}\] in $x$ has exactly two real distinct roots.($\sqrt t $ means the positive square root of $t$).