The length of the perpendicular from $(1, -2)$ to the line $12x + 5y + 63 = 0$ is

The line $L$ given by $\frac{x}{5}+\frac{y}{b}=1$ passes through the point $(13,32)$. The line $K$ is parallel to $L$ and has the equation $\frac{x}{c}+\frac{y}{3}=1$. Then the distance between $L$ and $K$ is

The line $L$ is given by $\frac{x}{5} + \frac{y}{b} = 1$ and passes through the point $(13, 32)$. The line $K$ is parallel to $L$ and has the equation $\frac{x}{c} + \frac{y}{3} = 1$. Then the distance between $L$ and $K$ is

Let the equation $x(x+2)(12-k)=2$ have equal roots. Then the distance of the point $(k, \frac{k}{2})$ from the line $3x+4y+5=0$ is

The length of the perpendicular from the point $(3, 1)$ to the line $4x + 3y + 20 = 0$ is:

Let $f(\theta)$ be the distance of the line $(\sqrt{\sin \theta})x + (\sqrt{\cos \theta})y + 1 = 0$ from the origin. Then the range of $f(\theta)$ is -