The equation of the straight line making equal intercepts on the axes and passing through the point $(2, 4)$ is

If a straight line through the point $P(1, 2)$,which makes an angle $45^{\circ}$ with the $X$-axis,meets the line $3x + 4y + 5 = 0$ at $Q$,then the length of $PQ$ equals ......... units.

If the line passing through the points $(4, 3)$ and $(2, \lambda)$ is perpendicular to the line $y = 2x + 3$,then find the value of $\lambda$.

The $y$-intercept made by the line $x \cos \alpha + y \sin \alpha = a$ is:

$A$ man is walking on a straight line. The arithmetic mean of the reciprocals of the intercepts of this line on the coordinate axes is $\frac{1}{4}$. Three stones $A, B$ and $C$ are placed at the points $(1,1), (2,2)$ and $(4,4)$ respectively. Which of these stones is/are on the path of the man?

The points $A (1, 3)$ and $C (5, 1)$ are the opposite vertices of a rectangle. The equation of the line passing through the other two vertices and having a gradient of $2$ is: