Find the fixed point through which the line $x(a + 2b) + y(a + 3b) = a + b$ always passes for all values of $a$ and $b$.

The locus of the mid-point of the segment intercepted between the axes by the variable line $x \cos \alpha + y \sin \alpha = p$,where $p$ is a constant,is

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 $P$.

$A$ variable line passing through $(l, m)$ intersects the coordinate axes at the points $A$ and $B$. If the line drawn parallel to $Y$-axis through $A$ and parallel to $X$-axis through $B$ meet at $P$,then the locus of $P$ is

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

The ends of a rod of length $l$ move on two mutually perpendicular lines. The locus of the point on the rod which divides it in the ratio $1 : 2$ is