It is well known that quadratic forms satisfy the Hasse principle (this is in fact the Hasse-Minkowski theorem): that is, for a given quadratic form $Q(x\_1, \cdots, x\_n)$ having rational coefficients, the equation $Q(\mathbf{x}) = 0$ has a rational solution if and only if it has a real solution and a $\mathbb{Q}\_p$ solution for every prime $p$. In other words, if and only if the equation has a solution in every completion of $\mathbb{Q}$.


Does this hold for equations of the shape


$$\displaystyle ax^2 + by^2 = cz^3, a,b,c \in \mathbb{Z}$$
or
$$\displaystyle ax^2 + by^3 + cz^3 = 0, a,b,c \in \mathbb{Z}?$$



Local-to-global principle for certain genus 0 curves
The former always has rational solutions. Let $x=tu$, $y=tv$, $z=tw$, then
$$ au^2 + bv^2 = c t w^3. $$
So simply choose any $u,v,w\in\mathbb Q^\*$ that you want, set
$$ t = \frac{au^2+bv^2}{cw^3}, $$
and you'll get a rational solution
$$ \left( \frac{(au^2+bv^2)u}{cw^3},
\frac{(au^2+bv^2)v}{cw^3}
\frac{(au^2+bv^2)}{cw^2}\right)$$
to your equation. (Similarly for the second equation, but I'll let you work it out.)


