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

$L \equiv 7x - y + 8 = 0$ is one of the diagonals of a square for which $(-4, 5)$ and $(3, 4)$ are two vertices. Find the coordinates of the two vertices lying on the diagonal $L = 0$.

Let $A(h, k)$,$B(1, 1)$,and $C(2, 1)$ be the vertices of a right-angled triangle with $AC$ as the hypotenuse. If the area of the triangle is $1$,then which of the following can be the set of values for $k$?

In an isosceles right-angled triangle,if the equation of the hypotenuse is $3x + 4y = 4$ and its opposite vertex is $(2, 2)$,then the slopes of the remaining two sides are:

Let a triangle be bounded by the lines $L_{1}: 2x + 5y = 10$,$L_{2}: -4x + 3y = 12$,and the line $L_{3}$,which passes through the point $P(2, 3)$,intersects $L_{2}$ at $A$ and $L_{1}$ at $B$. If the point $P$ divides the line segment $AB$ internally in the ratio $1:3$,then the area of the triangle is equal to

In a quadrilateral $ABCD$,which is not a trapezium,it is known that $\angle DAB = \angle ABC = 60^{\circ}$. Moreover,$\angle CAB = \angle CBD$. Then,