The equation to the locus of a point which moves so that its distance from the $x$-axis is always one-half its distance from the origin is:

The locus of all points that are at a distance greater than $2$ units from $(-3, 0)$ is:

If the tangents to the circle $x^2 + y^2 = a^2$ with slopes $\alpha$ and $\beta$ intersect at point $P$,and $\cot \alpha + \cot \beta = 0$,then the locus of $P$ is:

If the coordinates of a moving point $P$ are $(\cos \theta + \sin \theta, \sin \theta - \cos \theta)$,where $\theta$ is a parameter,find the locus of $P$.

If $A$ and $B$ are the points of intersection of the circle $x^2+y^2-8x=0$ and the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$,and a point $P$ moves on the line $2x-3y+4=0$,then the centroid of $\triangle PAB$ lies on the line:

$A$ stick of length $l$ rests against the floor and a wall of a room. If the stick begins to slide on the floor,then the locus of its middle point is