$A$ straight line passes through a fixed point $(h, k)$. The locus of the foot of the perpendicular drawn from the origin to this line is:

$A$ 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 :

Through a given point $P(a, b)$,a straight line is drawn to meet the axes at $Q$ and $R$. If the parallelogram $OQSR$ is completed,then the equation of the locus of $S$ is (given $O$ is the origin):

The equation of the locus of points which are equidistant from the points $(2,3)$ and $(4,5)$ is

$A(5,3), B(3,-2), C(2,-1)$ are three points. If $P(x,y)$ is a variable point such that the area of the quadrilateral $PABC$ is $10$ sq. units,then the locus of $P$ is

If the distance from a variable point $P$ to the point $(4, 3)$ is equal to the perpendicular distance from $P$ to the line $x + 2y - 1 = 0$,then the equation of the locus of the point $P$ is