Let $P(2, -3)$ and $Q(-2, 1)$ be the vertices of the $\Delta PQR$. If the centroid of $\Delta PQR$ lies on the line $2x + 3y = 1$, then the locus of $R$ is

The locus of the orthocentre of the triangle formed by the lines $(1+p) x-p y+p(1+p)=0$,$(1+q) x-q y+q(1+q)=0$,and $y=0$,where $p \neq q$,is

$A$ ray of light passing through the point $A(1, 2)$ is reflected at a point $B$ on the $x$-axis and then passes through $(5, 3)$. Then the equation of $AB$ is:

$A$ variable line $L$ passing through the origin cuts two parallel lines $x-y+10=0$ and $x-y+20=0$ at two points $A$ and $B$ respectively. If $P$ is a point on line $L$ such that $OA, OP, OB$ are in harmonic progression,then the locus of $P$ is

Every point $(x, y)$ on the curve $3x + 2y - 3xy = 0$ is the centroid of a triangle formed by the coordinate axes and a line $(L)$ intersecting both the coordinate axes. Then all such lines $(L)$

Let $b, d > 0$. The locus of all points $P(r, \theta)$ for which the line $OP$ (where $O$ is the origin) intersects the line $r \sin \theta = b$ at $Q$ such that $PQ = d$ is