The locus of the point of intersection of tangents to the circle $x = a \cos \theta, y = a \sin \theta$ at the points whose parametric angles differ by $\pi / 2$ is:

The locus of the point given by the equations $x = \frac{2at}{1 + t^2}$ and $y = \frac{a(1 - t^2)}{1 + t^2}$ for $-1 \le t \le 1$ is a

The number of circles touching the line $y - x = 0$ and the $y$-axis is

The equation $r \cos \theta = 2 a \sin^2 \theta$ represents the curve

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:

$A$ 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