If $m_1$ and $m_2$ $(m_1 > m_2)$ are the slopes of the lines which make an angle of $30^{\circ}$ with the line joining the points $(1, 2)$ and $(3, 4)$,then $\frac{m_1}{m_2} = $

The acute angle between two straight lines passing through the point $M(-6, -8)$ and the points in which the line segment $2x + y + 10 = 0$ enclosed between the coordinate axes is divided in the ratio $1 : 2 : 2$ in the direction from the point of its intersection with the $x$-axis to the point of intersection with the $y$-axis is:

If $\alpha, \beta$ $(\alpha > \beta)$ are two values of $k$ such that the equations $2x + (3 - 2k)y + (2k + 1) = 0$ and $kx + (k - 1)y - 4 = 0$ represent two perpendicular lines,then $\alpha^2 + 2\beta =$

Consider a triangle $ABC$ having the vertices $A(1,2)$,$B(\alpha, \beta)$,and $C(\gamma, \delta)$. The angles are $\angle ABC = \frac{\pi}{6}$ and $\angle BAC = \frac{2\pi}{3}$. If the points $B$ and $C$ lie on the line $y = x + 4$,then $\alpha^2 + \gamma^2$ is equal to:

If two lines $x+(a-1)y=1$ and $2x+a^2y=1$ $(a \in R-\{0,1\})$ are perpendicular,then the distance of their point of intersection from the origin is

The equations of the lines passing through the point $(3,2)$ and making an acute angle of $45^{\circ}$ with the line $x-2y-3=0$ are