The angle between the line joining the points $(1, -2)$ and $(3, 2)$ and the line $x + 2y - 7 = 0$ is

The lines $p(p^2 + 1)x - y + q = 0$ and $(p^2 + 1)^2x + (p^2 + 1)y + 2q = 0$ are perpendicular to a common line for which value of $p$?

The angle between the straight lines $3x + 4y + 9 = 0$ and $x - 7y - 22 = 0$ is

The slope of a line is double the slope of another line. If the tangent of the angle between them is $\frac{1}{3},$ find the slopes of the lines.

Two lines pass through the point $(2,3)$ and intersect each other at an angle of $60^{\circ}$. If the slope of one line is $2$,find the equations of the other line.

Let the angle between the lines $x-2y+3=0$ and $kx-y+2=0$ be $45^{\circ}$. If $k_1, k_2$ $(k_1 > k_2)$ are two distinct real values of $k$,then $k_1-2=$