Let the equations of two adjacent sides of a parallelogram $ABCD$ be $2x - 3y = -23$ and $5x + 4y = 23$. If the equation of its one diagonal $AC$ is $3x + 7y = 23$ and the distance of $A$ from the other diagonal is $d$,then $50d^2$ is equal to $........$.

Let the area of the triangle with vertices $A(1, \alpha)$,$B(\alpha, 0)$,and $C(0, \alpha)$ be $4 \text{ sq. units}$. If the points $(\alpha, -\alpha)$,$(-\alpha, \alpha)$,and $(\alpha^2, \beta)$ are collinear,then $\beta$ is equal to:

Suppose a triangle is formed by $x+y=10$ and the coordinate axes. Then the number of points $(x, y)$ where $x$ and $y$ are natural numbers,lying inside the triangle is

In a $\triangle ABC$,$2x+3y+1=0$ and $x+2y-2=0$ are the perpendicular bisectors of its sides $AB$ and $AC$ respectively. If $A=(3,2)$,then the equation of the side $BC$ is

One side of a square is inclined at an acute angle $\alpha$ with the positive $x-$axis,and one of its extremities is at the origin. If the remaining three vertices of the square lie above the $x-$axis and the side of a square is $4$,then the equation of the diagonal of the square which is not passing through the origin is

The two vertices of a triangle are $(2, -1)$ and $(3, 2)$,and the third vertex lies on the line $x + y = 5$. If the area of the triangle is $4$ square units,then the third vertex is: