If the points $(1, 1)$,$(-1, -1)$,and $(-\sqrt{3}, k)$ are the vertices of an equilateral triangle,then what is the value of $k$?

The area of the parallelogram formed by the lines $L_1 \equiv \lambda x+4 y+2=0$,$L_2 \equiv 3 x+4 y-3=0$,$L_3 \equiv 2 x+\mu y+6=0$,and $L_4 \equiv 2 x+y+3=0$,where $L_1$ is parallel to $L_2$ and $L_3$ is parallel to $L_4$,is

The incentre of the triangle formed by the lines $x+y=1$,$x=1$,and $y=1$ is

Let $\left(5, \frac{a}{4}\right)$ be the circumcenter of a triangle with vertices $A(a, -2)$,$B(a, 6)$,and $C\left(\frac{a}{4}, -2\right)$. Let $\alpha$ denote the circumradius,$\beta$ denote the area,and $\gamma$ denote the perimeter of the triangle. Then $\alpha + \beta + \gamma$ is

If the orthocentre of the triangle formed by the lines $y = x + 1$,$y = 4x - 8$,and $y = mx + c$ is at $(3, -1)$,then $m - c$ is :

The orthocentre of the triangle formed by the lines $4x - 7y + 10 = 0$,$x + y = 5$,and $7x + 4y = 15$ is