From a point $A(1, 0)$ on the circle $x^2+y^2-2x+2y+1=0$,a chord $AB$ is drawn and it is extended to a point $P$ such that $AP=3AB$. The equation of the locus of $P$ is

If $A(1, 1)$,$B(-1, 1)$,and $C(-1, -1)$ are three points and a point $P(x, y)$ moves such that $PA^2 = PB^2 + PC^2$,then the equation of the locus of $P$ is:

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:

Let the circle $x^{2}+y^{2}=4$ intersect the $x$-axis at the points $A(a,0), a>0$ and $B(b,0)$. Let $P(2 \cos \alpha, 2 \sin \alpha), 0 < \alpha < \frac{\pi}{2}$ and $Q(2 \cos \beta, 2 \sin \beta)$ be two points such that $(\alpha - \beta) = \frac{\pi}{2}$. Then the point of intersection of $AQ$ and $BP$ lies on:

The locus of the point of intersection of perpendicular tangents drawn to the circle $x^2+y^2=10$ is

$A$ stick of length $10$ units 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: