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

For any value of $\theta$,if the straight lines $x \sin \theta + (1 - \cos \theta) y = a \sin \theta$ and $x \sin \theta - (1 + \cos \theta) y + a \sin \theta = 0$ intersect at $P(\theta)$,then the locus of $P(\theta)$ is a

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:

Let the locus of the centre $(\alpha, \beta)$,$\beta > 0$,of the circle which touches the circle $x^{2} + (y - 1)^{2} = 1$ externally and also touches the $x$-axis be $L$. Then the area bounded by $L$ and the line $y = 4$ is.

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

If the distance of a variable point $P(x, y)$ from a point $A(2, -2)$ is twice the distance of $P$ from the $Y$-axis,then the equation of the locus of $P$ is: