<s>[INST] Lagrange Multiplier: Distance to the Origin Find the points on the curve $x^2+xy+y^2=2$ that are closest to the origin.
Is there a way to use Lagrange multipliers to answer this question? 
 [/INST] Without using calculus,
$$8=(2x+y)^2+3y^2$$
WLOG $\sqrt3y=2\sqrt2\sin t, 2x+y=2\sqrt2\cos t\iff x=?$
So, we need to minimize $$\left(\dfrac{2\sqrt2\sin t}{\sqrt3}\right)^2+\left(\dfrac{2\sqrt2\cos t-\dfrac{2\sqrt2\sin t}{\sqrt3}}{2}\right)^2$$
$$=\dfrac{8\sin^2t}3+\dfrac{8(\sqrt3\cos t-\sin t)^2}{12}$$
$$=\dfrac{8\sin^2t+3\cos^2t+\sin^2t-2\sqrt3\sin t\cos t}3$$
$$=\dfrac{9(1-\cos2t)+3(1+\cos2t)-2\sqrt3\sin2t}6$$
$$=\dfrac{12-2\sqrt3(\sqrt3\cos2t+\sin2t)}6$$
using $\cos2t=1-2\sin^2t=2\cos^2t-1$
and use $$-\sqrt{a^2+b^2}\le a\cos2t+b\sin2t\le\sqrt{a^2+b^2}$$
</s>