The locus of the centre of a circle which passes through the origin and cuts off a length of $4$ units from the line $x=3$ is

$A$ circle is drawn to cut a chord of length $2a$ units along the $X$-axis and to touch the $Y$-axis. The locus of the centre of the circle is

If the endpoints of the hypotenuse of a right-angled triangle are $(2, 0)$ and $(0, 2)$,find the locus of its third vertex.

Let a variable line passing through the centre of the circle $x^2+y^2-16x-4y=0$ meet the positive coordinate axes at points $A$ and $B$. Then the minimum value of $OA+OB$,where $O$ is the origin,is equal to

Tangents are drawn from the point $(17,7)$ to the circle $x^2+y^2=169$.
$STATEMENT-1$: The tangents are mutually perpendicular.
$STATEMENT-2$: The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is $x^2+y^2=338$.

If the sum of the distances from a variable point $P$ to the given points $A(1,0)$ and $B(0,1)$ is $2$,then the locus of $P$ is