  For every even number $n$, and every $n$-dimensional smooth hypersurface of
$\mathbb{P}^{n+1}$ of degree $d$, we compute the periods of all its
$\frac{n}{2}$-dimensional complete intersection algebraic cycles. Furthermore,
we determine the image of the given algebraic cycle under the cycle class map
inside the De Rham cohomology group of the corresponding hypersurface in terms
of its Griffiths basis and the polarization. As an application, we use this
information to address variational Hodge conjecture for a non complete
intersection algebraic cycle. We prove that the locus of general hypersurfaces
containing two linear cycles whose intersection is of dimension less than
$\frac{n}{2}-\frac{d}{d-2}$, corresponds to the Hodge locus of any integral
combination of such linear cycles.
