Two equal sides of an isosceles triangle are along $-x+2y=4$ and $x+y=4$. If $m$ is the slope of its third side,then the sum of all possible distinct values of $m$ is:

What is the angle in degrees between the lines $y = x + 5$ and $y = \sqrt{3}x - 4$ (in $^\circ$)?

$A$ line $L$ passes through the point $(3, -2)$ and is inclined at an angle of $60^{\circ}$ to the line $\sqrt{3}x + y = 1$. If $L$ also intersects the $x$-axis,then the equation of $L$ 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:

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 :

If $m=1$ is the slope of a line $L$,then the product of the slopes of non-parallel lines which are inclined at an angle of $60^{\circ}$ with $L$ is