The image of every point lying on the curve $x^2+y^2=1$ in the line $x+y=1$ satisfies the equation:

$A$ point $P(x, y)$ is such that its distances from $(-1, 0)$ and $(0, 2)$ are in the ratio $\sqrt{2} : 1$. Then the locus of $P$ is

If the locus of the mid-point of the line segment from the point $(3, 2)$ to a point on the circle $x^{2} + y^{2} = 1$ is a circle of radius $r$,then $r$ is equal to:

The locus of the third vertex of a right-angled triangle,the ends of whose hypotenuse are $(1,2)$ and $(4,5)$,is

$A$ variable circle is drawn passing through the origin $O$. It intersects the $X$ and $Y$ axes at points $A$ and $B$ respectively,such that $OA + 2OB = K$ (a non-zero constant). The circle always passes through a fixed point $P$ other than the origin. The point $P$ lies on -

The locus of the midpoint of the chords of the circle $x^2 + y^2 - 2x - 2y - 2 = 0$ which makes an angle of $120^\circ$ at the centre is