$A$ circle passing through the point $P(\alpha, \beta)$ in the first quadrant touches the two coordinate axes at the points $A$ and $B$. The point $P$ is above the line $AB$. The point $Q$ on the line segment $AB$ is the foot of the perpendicular from $P$ on $AB$. If $PQ$ is equal to $11$ units,then the value of $\alpha \beta$ is $.............$.

If $A\left(\frac{\pi}{3}\right)$ and $B\left(\frac{\pi}{6}\right)$ are points on a circle represented in parametric form with center $(0,0)$ and radius $12$,then the length of the chord $AB$ is:

The number of common tangents to the circles ${x^2} + {y^2} - x = 0$ and ${x^2} + {y^2} + x = 0$ is:

$A$ circular wire of radius $7\,cm$ is cut and bent again into an arc of a circle of radius $12\,cm$. The angle subtended by the arc at the centre is ......$^o$

Let $C_1$ and $C_2$ be two circles touching each other externally at point $A$. Let $AB$ be the diameter of circle $C_1$. Draw a secant $BA_3$ to circle $C_2$,intersecting circle $C_1$ at point $A_1$ (where $A_1 \neq A$) and circle $C_2$ at points $A_2$ and $A_3$. If $BA_1 = 2$,$BA_2 = 3$,and $BA_3 = 4$,then the radii of circles $C_1$ and $C_2$ are respectively:

In a circle with center $O$,suppose $A, P, B$ are three points on its circumference such that $P$ is the mid-point of the minor arc $AB$. Suppose when $\angle AOB = \theta$,$\frac{\text{area}(\triangle AOB)}{\text{area}(\triangle APB)} = \sqrt{5} + 2$. If $\angle AOB$ is doubled to $2\theta$,then the ratio $\frac{\text{area}(\triangle AOB)}{\text{area}(\triangle APB)}$ is