$A$ variable chord is drawn through the origin to the circle $x^2 + y^2 - 2ax = 0$. The locus of the centre of the circle drawn on this chord as diameter is:

$A$ piece of paper in the shape of a sector of a circle (see $Fig. 1$) is rolled up to form a right-circular cone (see $Fig. 2$). The value of the angle $\theta$ is

$A$ cow is tied to a corner (vertex) of a regular hexagonal fenced area of side $a \ m$ by a rope of length $\frac{5a}{2} \ m$ in a grass field. (The cow cannot graze inside the fenced area). What is the maximum possible area of the grass field to which the cow has access to graze?

The equation to the locus of a point which moves so that its distance from the $x$-axis is always one-half its distance from the origin 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:

$A$ company situated at $(2,0)$ in the $XY$-plane charges $RS. 2$ per $km$ for delivery. $A$ second company at $(0,3)$ charges $RS. 3$ per $km$ for delivery. The region of the plane where it is cheaper to use the first company is