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 center of a circle which touches the circle $x^{2} + y^{2} - 6x - 6y + 14 = 0$ externally and also touches the $y$-axis is:

The maximum possible value of $x^2+y^2-4x-6y$,where $x, y$ are real,subject to the condition $|x+y|+|x-y|=4$ is

$A$ bottle in the shape of a right-circular cone with height $h$ contains some water. When its base is placed on a flat surface,the height of the vertex from the water level is $a$ units. When it is kept upside down,the height of the base from the water level is $\frac{a}{4}$ units. Then the ratio $\frac{h}{a}$ is

Two line segments $AB$ and $CD$ are constrained to move along the $X$ and $Y$-axes,respectively,in such a way that the points $A, B, C, D$ are concyclic. If $AB = a$ and $CD = b$,then the locus of the centre of the circle passing through $A, B, C, D$ in polar coordinates is

If the locus of the mid-point of the line segment from the point $(3, 2)$ to a point on the circle $x^{2} + y^{2} = 1$ is a circle of radius $r$,then $r$ is equal to: