If the shortest distance between the lines $\bar{r}_1=\alpha \hat{i}+2 \hat{j}+2 \hat{k}+\lambda(\hat{i}-2 \hat{j}+2 \hat{k}), \lambda \in \mathbb{R}, \alpha>0 \quad$ and $\bar{r}_2=-4 \hat{i}-\hat{k}+\mu(3 \hat{i}-2 \hat{j}-2 \hat{k}), \mu \in R$, is 9 , then the value of $\alpha$ is

If two curves $x^2-4 y^2=2$ and $8 x^2=40-\mathrm{m} y^2$ are orthogonal to each other then $\mathrm{m}=$

The resultant of two vectors $\vec{A}$ and $\vec{B}$ is $\vec{C}$. If the magnitude of $\vec{B}$ is doubled, the new resultant vector becomes perpendicular to $\vec{A}$, then the magnitude of $\overrightarrow{\mathrm{C}}$ is

A convex lens of focal length $\frac{1}{3} \mathrm{~m}$ forms a real, inverted image twice the size of the object. The distance of the object from the lens is