Let $q(x,y,z) = ax^2 + by^2 + cz^2$ be a non-singular diagonal ternary quadratic form with integer coefficients. The discriminant $\Delta(q)$ of $q$ is then equal to $abc$, and for any positive number $X$ there are finitely many non-singular diagonal quadratic forms of discriminant at most $X$. Indeed, the number of non-singular diagonal ternary quadratic forms with $|\Delta(q)| \leq X$ is approximated by the integral

$$\displaystyle 8\iiint_{\substack{1 \leq uvw \leq X \\ u,v,w \geq1}} dudvdw = 8X (\log X)^2 + O(X \log X).$$

A quadratic form $q \in \mathbb{Z}[x,y,z]$ (not necessarily diagonal) is *isotropic* if the equation

$$\displaystyle q(x,y,z) = 0$$

has a solution $(x,y,z) \in \mathbb{Z}^3$. Define

$$\displaystyle N(X) = \#\{q = ax^2 + by^2 + cz^2 : a,b,c \in \mathbb{Z} \setminus \{0\}, |\Delta(q)| \leq X, q \text{ is isotropic}\}.$$

Define

$$\displaystyle R(X) = \frac{X (\log X)^2}{N(X)}.$$

Is there an asymptotic relation for $R(X)$? Alternatively, how dense are isotropic diagonal ternary quadratic forms among all diagonal ternary quadratic forms?