$A$ point equidistant from the points $(2, 0)$ and $(0, 2)$ is

Consider the triangle formed by the lines $x + y = 0$,$x - y = 0$,and $lx + my = 1$. If $l$ and $m$ vary subject to the condition $l^2 + m^2 = 1$,then the locus of its circumcentre is:

If $A=(-1, 2)$ and $B=(1, -2)$ are two points and $P$ is a variable point such that the area of $\triangle PAB$ is always $1$,then the equation of the locus of $P$ is

The Cartesian form of the polar equation $\theta = \tan^{-1} 2$ 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

Let $A(5, -3)$,$B(3, -2)$,and $C(-1, 5)$ be three points. If $P$ is a point satisfying the condition $PA^2 + 2PB^2 = 3PC^2$,then a point that lies on the locus of $P$ is