If a line $AB$ of length $r$ moves so that $A$ and $B$ always lie respectively on the $X$-axis and the line $y=6x$,then the locus of the mid-point of $AB$ is:

Let the algebraic sum of the perpendicular distances from the points $A(2, 0)$,$B(0, 2)$,and $C(1, 1)$ to a variable line be zero. Then,all such lines:

Let $A = (a, 0)$ and $B = (-a, 0)$ be two fixed points. For $a \in (-\infty, 0)$,point $P(x, y)$ moves in the plane such that $PA = nPB$ $(n \neq 0, 1)$. If $n = 1$,find the locus of point $P$.

Let $A$ be the point $(0,4)$ in the $xy$-plane and let $B$ be the point $(2t, 0)$. Let $L$ be the midpoint of $AB$ and let the perpendicular bisector of $AB$ meet the $y$-axis at $M$. Let $N$ be the midpoint of $LM$. Then the locus of $N$ is

If the algebraic sum of the distances from the points $(2,0)$, $(0,2)$, and $(1,1)$ to a variable straight line is zero, then the line passes through the fixed point:

If $M$ is the foot of the perpendicular drawn from the origin $O$ to a variable line $L$ passing through a fixed point $Q(a, b)$,then the locus of the mid-point of $OM$ is