Distribution of lattice points on hyperboloids

Consider the region $\Omega_0$ on the hyperboloid $1=b^2+ac$ defined by the conditions
$$ 0<L_1\le a\le L_2<1,\quad 0<t_1\le\frac ba\le t_2<1. $$
Let $r(n,\Omega_0)_pr$ be the number of integral points $(a,b,c)$ with $a=p$ (a prime) on the hyperboloid $n=b^2+ac$ ($n>0$ is an integer) such that $(a,b,c)/\sqrt n\in\Omega_0$. It is proved that for prime $P>P(\varepsilon)$, $\varepsilon>0$,
$$ (K-\Delta-\varepsilon)\frac P{\log P}\le r(P^2,\Omega_0)_{pr}\le(K+\Delta+\varepsilon) \frac P{\log P}, $$
where
$$ K=2(t_2-t_1)(L_2-L_1),\quad\Delta=L^2_2\cdot\frac{2\pi}3. $$