If a bar of given length moves with its extremities on two fixed straight lines at right angles,then the locus of any point marked on the bar describes a/an

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 side $AB$ of $\triangle ABC$ is fixed and is of length $2a$ units. The vertex $C$ moves in the plane such that the vertical angle $\angle ACB$ is always constant and is equal to $\alpha$. Let the $x$-axis be along $AB$ and the origin be at $A$. Then the locus of the vertex $C$ is:

Any circle passing through the point of intersection of the lines $x + \sqrt{3}y = 1$ and $\sqrt{3}x - y = 2$ intersects these lines at points $P$ and $Q$. The angle subtended by the arc $PQ$ at its centre is- ............. $^o$

If $A(5, -4)$ and $B(7, 6)$ are points in a plane,then the set of all points $P(x, y)$ in the plane such that $AP:PB = 2:3$ is

The locus of the centre of the circle,which cuts the circle $x^2+y^2-20x+4=0$ orthogonally and touches the line $x=2$,is