Let the circle with centre at origin pass through the vertices of an equilateral triangle ABC . If $A \equiv(2,4)$, then the length of the median through A is

Let $\bar{a}=\hat{i}+\hat{j}+\hat{k}, \bar{b}$ and $\bar{c}=\hat{j}-\hat{k}$ be three vectors such that $\overline{\mathrm{a}} \times \overline{\mathrm{b}}=\overline{\mathrm{c}}$ and $\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=1$. If the length of projection vector of the vector $\overline{\mathrm{b}}$ on the vector $\overline{\mathrm{a}} \times \overline{\mathrm{c}}$ is $l$, then the value of $3 l^2$ is

The distance of the point $(1,2)$ from the line $x+y=0$ measured parallel to the line $3 x-y=2$ is

$$ \mathop {\lim }\limits_{x \to 2} \frac{x+3 x^2+5 x^3+7 x^4-166}{x-2}= $$