If the sum of the distances of a point from two perpendicular lines in a plane is $1$,find its locus.

The set of all points that forms a triangle of area $15$ sq units with the points $(1, -2)$ and $(-5, 3)$ lies on

If $M$ is a point on the line $y=x$ and points $P(0,1), Q(2,0)$ are such that $PM+QM$ is minimum,the coordinates of $M$ are

The locus of points $(x, y)$ in the plane satisfying $\sin ^2 x + \sin ^2 y = 1$ consists of

$A$ straight rod of length $4$ units slides such that its ends $A$ and $B$ always lie on the $X$ and $Y$-axes respectively. Then,the locus of the centroid of $\triangle OAB$ is

$p, x_1, x_2, \ldots, x_n$ and $q, y_1, y_2, \ldots, y_n$ are two arithmetic progressions with common differences $a$ and $b$ respectively. If $\alpha$ and $\beta$ are the arithmetic means of $x_1, x_2, \ldots, x_n$ and $y_1, y_2, \ldots, y_n$ respectively,then the locus of $P(\alpha, \beta)$ is