Distance bound from two hyperplane constraints ($|x|geq|A-B|/|a-b|$)

If $a\ne b$ and $x\cdot a=A$, $x\cdot b=B$, the Cauchy-Schwarz inequality applied to $x\cdot (a-b)=A-B$ gives $\Vert x\Vert \ge |A-B|/\Vert a-b\Vert$. Equality requires $x$ parallel to $a-b$ and satisfying both original equations. The estimate gives a lower bound on the distance from the origin to the intersection; it need not be sharp if that equality condition is incompatible with the constraints.