Track / Stage 4 / 251 of 340 #511 of 1964

Problem 511

AMC 12 late, AIME early

Algebra Difficulty 4.9 Find the answer

2. Given vectors $\vec{a}=\{1, \sqrt{2}\}, \vec{b}=\{-\sqrt{2}, 1\}$ . If positive numbers $k$ and $t$ make
$\vec{x}=\vec{a}+\left(t^{2}+1\right) \vec{b} \text { and } \vec{y}=-k \vec{a}+\frac{1}{t} \vec{b}$

perpendicular, then the minimum value of $k$ is

Your answer

A number or a short expression. Spacing, $ signs and \frac vs / are all fine.

Official solution

$2.2$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.