If a chord is drawn from the origin to the circle $(x - 1)^2 + y^2 = 1$,then the equation of the locus of the midpoint of this chord is:

The locus of the centre of the circle which cuts off intercepts of length $2a$ and $2b$ from the $x$-axis and $y$-axis respectively,is

The perimeter of the locus of the point $P$ which divides the line segment $QA$ internally in the ratio $1:2$,where $A=(4,4)$ and $Q$ lies on the circle $x^2+y^2=9$ is

Let the locus of the midpoints of the chords of the circle $x^2+(y-1)^2=1$ drawn from the origin intersect the line $x+y=1$ at $P$ and $Q$. Then,the length of $PQ$ is:

The angle between a pair of tangents drawn from a point $P$ to the circle ${x^2} + {y^2} + 4x - 6y + 9{\sin ^2}\alpha + 13{\cos ^2}\alpha = 0$ is $2\alpha$. The equation of the locus of the point $P$ is

The locus of the centre of a circle which passes through the point $(a, 0)$ and touches the line $x + 1 = 0$ is