  A recent series of papers has examined the extension of
disjunctive-programming techniques to mixed-integer second-order-cone
programming. For example, it has been shown---by several authors using
different techniques---that the convex hull of the intersection of an
ellipsoid, $E$, and a split disjunction, $(l - x_j)(x_j - u) \le 0$ with $l <
u$, equals the intersection of $E$ with an additional second-order-cone
representable (SOCr) set. In this paper, we study more general intersections of
the form $K \cap Q$ and $K \cap Q \cap H$, where $K$ is a SOCr cone, $Q$ is a
nonconvex cone defined by a single homogeneous quadratic, and $H$ is an affine
hyperplane. Under several easy-to-verify conditions, we derive simple,
computable convex relaxations $K \cap S$ and $K \cap S \cap H$, where $S$ is a
SOCr cone. Under further conditions, we prove that these two sets capture
precisely the corresponding conic/convex hulls. Our approach unifies and
extends previous results, and we illustrate its applicability and generality
with many examples.
