Distance of ellipse to the origin
Solution 1:
You can always try to write it in polar coordinates:
$$3r^2+4r^2\sin\theta\cos\theta=20$$ $$r^2=\frac{20}{3+2\sin2\theta}\ge\frac{20}{3+2}=4$$
Solution 2:
Hint: $4xy \le 2(x^2+y^2), 3x^2+3y^2 = 3(x^2+y^2) \implies 5(x^2+y^2) \ge 20 \implies x^2+y^2 \ge .....$