If the point $(a, a)$ lies between the lines $|x + y| = 2$,then

The length of the perpendicular from the point $(a \cos \alpha, a \sin \alpha)$ upon the straight line $y = x \tan \alpha + c$,where $c > 0$,is:

If $\alpha = 1 + \sum_{r=1}^6 (-3)^{r-1} \binom{12}{2r-1}$,then the distance of the point $(12, \sqrt{3})$ from the line $\alpha x - \sqrt{3} y + 1 = 0$ is ..........

$A$ straight line passes through the points $(5,0)$ and $(0,3)$. The length of the perpendicular from the point $(4,4)$ to the line is:

Let $A$ be the point of intersection of the lines $3x + 2y = 14$ and $5x - y = 6$. Let $B$ be the point of intersection of the lines $4x + 3y = 8$ and $6x + y = 5$. The distance of the point $P(5, -2)$ from the line $AB$ is:

Let the expression $E = 8^a + 8^b - 3 \cdot 2^{a+b}$ take its minimum value $p$ at $a = \alpha$ and $b = \beta$. Then,the perpendicular distance of the point $P(\alpha, \beta)$ from the line $x + y + 2p = 0$ is