If the line $lx + my = 1$ is a tangent to the circle ${x^2} + {y^2} = {a^2}$,then the locus of the point $(l, m)$ is

The locus of a point which moves so that its distance from the $x$-axis is double its distance from the $y$-axis 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

The side $AB$ of $\triangle ABC$ is fixed and is of length $2a$ units. The vertex $C$ moves in the plane such that the vertical angle $\angle ACB$ is always constant and is equal to $\alpha$. Let the $x$-axis be along $AB$ and the origin be at $A$. Then the locus of the vertex $C$ is:

Without changing the direction of the coordinate axes,the origin is shifted to $(h, k)$ such that the linear (first-degree) terms in the equation $x^2 + y^2 - 4x + 6y - 7 = 0$ are eliminated. Then the point $(h, k)$ is:

$A$ stick of length $l$ 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