$A$ rod of length $8$ units moves such that its ends $A$ and $B$ always lie on the lines $x-y+2=0$ and $y+2=0$,respectively. If the locus of the point $P$,that divides the rod $AB$ internally in the ratio $2:1$ is $9(x^2+\alpha y^2+\beta xy+\gamma x+28y)-76=0$,then $\alpha-\beta-\gamma$ is equal to :

If two mutually perpendicular lines through the point $A(1, 1)$ intersect the $x$-axis and $y$-axis at points $B$ and $C$ respectively,then the locus of the centroid of $\Delta ABC$ is -

$A$ $10 \text{ inches}$ long pencil $AB$ with midpoint $C$ and a small eraser $P$ are placed on the horizontal top of a table such that $PC = \sqrt{5} \text{ inches}$ and $\angle PCB = \tan^{-1}(2)$. The acute angle through which the pencil must be rotated about $C$ so that the perpendicular distance between the eraser and the pencil becomes exactly $1 \text{ inch}$ is:

$A \equiv (\cos \theta, \sin \theta)$ and $B \equiv (\sin \theta, -\cos \theta)$ are two points. The locus of the centroid of $\triangle OAB$,where $O$ is the origin,is

$A$ point $P$ on a line is at a distance of $4$ units from the origin $(0,0)$. If the line makes an angle of $60^{\circ}$ with the negative direction of the $X$-axis,then the coordinates of $P$ are

For the variable $t$, the locus of the point of intersection of the lines $3tx - 2y + 6t = 0$ and $3x + 2ty - 6 = 0$ is