The acute angle between the lines $x-y=0$ and $y=0$ is

The acute angle between the line $4x - 2y + 13 = 0$ and the line which makes equal intercepts with the coordinate axes is

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 the line through the points $(-2, 6)$ and $(4, 8)$ is perpendicular to the line passing through the points $(8, 12)$ and $(x, 24)$,then the value of $x$ is

The equation of the straight line passing through the point $(3, 2)$ and inclined at an angle of $60^{\circ}$ with the line $\sqrt{3} x + y = 1$ is

The slope of a line $L$ is $2$. If $m_1$ and $m_2$ are the slopes of two lines which are inclined at an angle of $\frac{\pi}{6}$ with $L$,then $m_1 + m_2 =$