We solve in Rⁿ the problem of distance evaluation between a quadric and a manifold obtained as the intersection of another quadric and a linear manifold. Application of Elimination Theory algorithms for the system of algebraic equations of the Lagrange multipliers method results in construction of the distance equation, i.e., a univariate algebraic equation one of the zeros of which (generically minimal positive) coincides with the square of the distance between considered manifolds. We also deduce the necessary and sufficient algebraic conditions under which the manifolds intersect and propose an algorithm for finding the coordinates of their nearest points.