Let $C$ be a circle with radius $3$ and center at $(0, 0)$. The equation of the locus of the midpoint of the chord of the circle that subtends an angle of $\frac{2\pi}{3}$ at the center is:

If the line $lx + my = 1$ is a tangent to the circle ${x^2} + {y^2} = {a^2}$,then the locus of the point $(l, m)$ is

The equation of the locus of a point whose distance from $(a, 0)$ is equal to its distance from the $y$-axis is

Let $A(5,4)$ and $B(5,-4)$ be two points. If $P(x,y)$ is a point in the coordinate plane such that $\angle APB = \frac{\pi}{4}$,then the point $P$ lies on the curve

$A(a, 0)$ is a fixed point and $\theta$ is a parameter such that $0 < \theta < 2 \pi$. If $P(a \cos \theta, a \sin \theta)$ is a point on the circle $x^2+y^2=a^2$ and $Q(b \sin \theta, -b \cos \theta)$ is a point on the circle $x^2+y^2=b^2$,then the locus of the centroid of the triangle $APQ$ is

For two points $A(2,1)$ and $B(1,2)$,$P$ is a point such that $PA:PB = 2:1$. Then the locus of $P$ is: