A point moves such that its distance from the point $(4,\,0)$is half that of its distance from the line $x = 16$. The locus of this point is

One side of a rectangle lies along the line $4x + 7y + 5 = 0.$ Two of its vertices are $(-3, 1)$ and $(1, 1)$. Then the equations of other three sides are

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

Two sides of a parallelogram are along the lines, $x + y = 3$ and $x -y + 3 = 0$. If its diagonals intersect at $(2, 4)$, then one of its vertex is

Let two points be $\mathrm{A}(1,-1)$ and $\mathrm{B}(0,2) .$ If a point $\mathrm{P}\left(\mathrm{x}^{\prime}, \mathrm{y}^{\prime}\right)$ be such that the area of $\Delta \mathrm{PAB}=5\; \mathrm{sq}$ units and it lies on the line, $3 x+y-4 \lambda=0$ then a value of $\lambda$ is

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