If $A(5, -4)$ and $B(7, 6)$ are points in a plane,then the set of all points $P(x, y)$ in the plane such that $AP:PB = 2:3$ is

The locus of the centre of the circle which cuts a chord of length $2a$ from the positive $x$-axis and passes through a point on the positive $y$-axis at a distance $b$ from the origin is:

The coordinates of the points $A$ and $B$ are $(ak, 0)$ and $(\frac{a}{k}, 0)$,where $k = \pm 1$. If a point $P(x, y)$ moves such that $PA = kPB$,then the equation to the locus of $P$ is:

The locus of the centre of a circle passing through $(a, b)$ and cutting orthogonally to the circle $x^2 + y^2 = p^2$ is

The locus of a point which moves such that the sum of the squares of its distances from the sides of a square of side length $1$ is $9$,is

The locus of the midpoints of the chords of the circle $C_1: (x-4)^2 + (y-5)^2 = 4$ which subtend an angle $\theta_i$ at the centre of the circle $C_1$,is a circle of radius $r_i$. If $\theta_1 = \frac{\pi}{3}$,$\theta_3 = \frac{2\pi}{3}$ and $r_1^2 = r_2^2 + r_3^2$,then $\theta_2$ is equal to