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

Let $\mathrm{A}(-2,-1), \mathrm{B}(1,0), \mathrm{C}(\alpha, \beta)$ and $\mathrm{D}(\gamma, \delta)$ be the vertices of a parallelogram $A B C D$. If the point $C$ lies on $2 x-y=5$ and the point $D$ lies on $3 x-2 y=6$, then the value of $|\alpha+\beta+\gamma+\delta|$ is equal to_____.

The diagonals of the parallelogram whose sides are $lx + my + n = 0,$ $lx + my + n' = 0$,$mx + ly + n = 0$, $mx + ly + n' = 0$ include an angle

The line $2x + 3y = 12$ meets the $x -$ axis at $A$ and the $y -$ axis at $B$ . The line through $(5, 5)$ perpendicular to $AB$ meets the $x -$ axis, $y -$ axis $\&$ the line $AB$ at $C, D, E$ respectively. If $O$ is the origin, then the area of the $OCEB$ is :

The equation of straight line passing through $( - a,\;0)$ and making the triangle with axes of area ‘$T$’ is

Two lines are drawn through $(3, 4)$, each of which makes angle of $45^\circ$ with the line $x - y = 2$, then area of the triangle formed by these lines is