The number of rational points on a class of hypersurfaces in quadratic extensions of finite fields

Let $q$ be an even prime power and let ${\mathbb{F}}_q$ be the finite field of $q$ elements. Let $f$ be a nonzero polynomial over ${\mathbb{F}}_{q^2}$ of the form $f=a_1{x}_1^{m_1}+\cdots +a_s{x}_s^{m_s}+y_1{y}_2+\cdots +y_{n-1}{y}_n+y_{n-2t-1}^2+\cdots +y_{n-3}^2+y_{n-1}^2$ $+b_t{y}_{n-2t}^2+$ $\cdots +b_1{y}_{n-2}^2+b_0{y}_n^2$, where $a_i,b_j\in {\mathbb{F}}_{q^2}^{\ast },$ $m_i\ne 1,$ $(m_i,m_k)=1,$ $i\ne k,$ $m_i|(q+1),$ $m_i\in {\mathbb{Z}}^{+},$ $2|n$, $n>2$, $0\le t\le \frac{n}{2}-2$, ${\mathrm{T}\mathrm{r}}_{{\mathbb{F}}_{q^2}/{\mathbb{F}}_2}(b_j)=1$ for $i,k=1,\dots ,s$ and $j=0,1,\dots ,t$. For each $b\in {\mathbb{F}}_{q^2}$, let $N_{q^2}(f=b)$ denote the number of ${\mathbb{F}}_{q^2}$-rational points on the affine hypersurface $f=b$. In this paper, we obtain the formula of $N_{q^2}(f=b)$ by using the Jacobi sums, Gauss sums and the results of quadratic form in finite fields.