The point whose abscissa is equal to its ordinate and which is equidistant from the points $(1, 0)$ and $(0, 3)$ is

$A$ straight line passing through a fixed point $(2, 3)$ intersects the coordinate axes at distinct points $P$ and $Q$. If $O$ is the origin and the rectangle $OPRQ$ is completed,then the locus of $R$ is:

$A$ straight line moves such that the sum of the reciprocals of its intercepts on two perpendicular lines is constant. Then the line always passes through:

If a variable straight line passing through the point of intersection of the lines $x-2y+3=0$ and $2x-y-1=0$ intersects the $X, Y$-axes at $A$ and $B$ respectively,then the equation of the locus of a point which divides the segment $AB$ in the ratio $-2:3$ is

If a point $P$ is equidistant from the points $A(a + b, b - a)$ and $B(a - b, a + b)$,find the locus of $P$.

If $A(2,-3)$ and $B(-2,1)$ are two vertices of a triangle and the third vertex moves on the line $2x + 3y = 9$,then the locus of the centroid of the triangle is