Distance of ellipse to the origin

You can always try to write it in polar coordinates:

$$3r^2+4r^2\sin\theta\cos\theta=20$$ $$r^2=\frac{20}{3+2\sin2\theta}\ge\frac{20}{3+2}=4$$

Hint: $4xy \le 2(x^2+y^2), 3x^2+3y^2 = 3(x^2+y^2) \implies 5(x^2+y^2) \ge 20 \implies x^2+y^2 \ge .....$

This ellipse is centered at the orgin.

The distance you seek equals the length of the minor axis.

Next thing to know is that it has been rotate 45 degrees.

$x = \frac {\sqrt 2}{2} x' + \frac {\sqrt 2}{2} y'\\ y = -\frac {\sqrt 2}{2} x' + \frac {\sqrt 2}{2} y'$

Will rotate it into standard position.

Or, the egienvalues of $\begin{bmatrix} 3&2\\2&3\end{bmatrix}$ are $1$ and $5$

$x'^2 + 5y'^2 = 20\\ \frac {x'^2}{20} + \frac {y'^2}{4} = 1$

$2$