On a problem related to second-order Diophantine equations

The article considers the problem set by V. N. Ushakov of finding triangles with integer lengths of sides $a$, $b$, $c$, satisfying the relations $a^2=b^2+c^2+k$ and $\dfrac{a}{c}=\dfrac{3}{2}$, where $k$ is a nonzero integer. We give a necessary and sufficient condition for the number $k$ under which such triangles exist. The proof is constructive and allows, in the case of satisfying the criterion, to indicate an infinite number of triples $(a,b,c)$ with the given property.