Let the locus of the centre $(\alpha, \beta)$,$\beta > 0$,of the circle which touches the circle $x^{2} + (y - 1)^{2} = 1$ externally and also touches the $x$-axis be $L$. Then the area bounded by $L$ and the line $y = 4$ is.

The locus of the centre of circles passing through $(a, b)$ and cutting the circle $x^2+y^2-2x+4y-4=0$ orthogonally is

$A$ line through $(0,0)$ cuts the circle $x^2 + y^2 - 2ax = 0$ at points $A$ and $B$. The locus of the centre of the circle drawn on $AB$ as a diameter 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

If the distance from a variable point $P$ to a fixed point $A(a, 0)$ is equal to the perpendicular distance from $P$ to the line $x+y=0$,then the equation of the locus of $P$ is

The locus of a point which is at a distance of $4$ units from $(3, -2)$ in the $xy$-plane is