$A$ point moves in such a way that its distance from the origin is always $4$. Then the locus of the point 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

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:

Let the point $(p, p+1)$ lie inside the region $E = \{(x, y) : 3-x \leq y \leq \sqrt{9-x^2}, 0 \leq x \leq 3\}$. If the set of all values of $p$ is the interval $(a, b)$,then $b^2+b-a^2$ is equal to $.................$.

If the tangents drawn from a point $P$ to the circle $x^2 + y^2 = a^2$ are perpendicular to the tangents drawn to the circle $x^2 + y^2 = b^2$,then the locus of $P$ is:

If the angle between a pair of tangents drawn from a point $P$ to the circle $x^2+y^2-4x+2y+3=0$ is $\frac{\pi}{2}$,then the locus of $P$ is