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 $k_1-2=$

Consider the lines $L_1: x - y = 1,$ $L_2: x + y = 1,$ $L_3: 2x + 2y = 5,$ and $L_4: 2x - 2y = 7.$ Which of the following statements is true?

If the equation of a line having a slope $m$ $(m \in \mathbb{Z})$,passing through $(1, 1)$ and making an angle of $\tan^{-1}\left(\frac{5}{7}\right)$ with the line $x + y - 3 = 0$ is $ax + y + c = 0$,then $ac =$

Show that two lines $a_{1}x + b_{1}y + c_{1} = 0$ and $a_{2}x + b_{2}y + c_{2} = 0$,where $b_{1}, b_{2} \neq 0$,are perpendicular if $a_{1}a_{2} + b_{1}b_{2} = 0$.

The angle between the lines $2x + 11y - 7 = 0$ and $x + 3y + 5 = 0$ is

The acute angle between the lines $x \sin \theta - y \cos \theta = 5$ and $x \sin \alpha - y \cos \alpha + 11 = 0$ is