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 the mid-points of the chords of the circle $x^2+y^2=16$ which are tangents to the hyperbola $9x^2-16y^2=144$ is

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:

The locus of the centres of the circles,which touch the circle $x^2 + y^2 = 1$ externally,also touch the $y$-axis and lie in the first quadrant is

Find the locus of the midpoint of the chord of the circle $x^2 + y^2 = 16$ which is a tangent to the hyperbola $9x^2 - 16y^2 = 144$.

Let $P$ be a variable point on a circle $C$ and $Q$ be a fixed point outside $C$. If $R$ is the midpoint of the line segment $PQ$, then the locus of $R$ is