Rational points on certain hyperelliptic curves over finite fields

Let $K$ be a field, $a, b\in K$ and $ab\neq 0$. Let us consider the polynomials $g_{1}(x)=x^n+ax+b, g_{2}(x)=x^n+ax^2+bx$, where $n$ is a fixed positive integer. In this paper we show that for each $k\geq 2$ the hypersurface given by the equation \begin{equation*} S_{k}^{i}: u^2=\prod_{j=1}^{k}g_{i}(x_{j}),\quad i=1, 2. \end{equation*} contains a rational curve. Using the above and Woestijne's recent results \cite{Woe} we show how one can construct a rational point different from the point at infinity on the curves $C_{i}:y^2=g_{i}(x), (i=1, 2)$ defined over a finite field, in polynomial time.