Small Zeros of Quadratic Forms with Linear Conditions

Given a quadratic form and $M$ linear forms in $N+1$ variables with coefficients in a number field $K$, suppose that there exists a point in $K^{N+1}$ at which the quadratic form vanishes and all the linear forms do not. Then we show that there exists a point like this of relatively small height. This generalizes a result of D.W. Masser (1998). As a corollary of this result, we prove an extension of Cassels' theorem on small zeros of quadratic forms (1955) to non-singular small zeros over a number field.