The tangents are drawn from the point $(4, 5)$ to the circle $x^2 + y^2 - 4x - 2y - 11 = 0$. The area of the quadrilateral formed by these tangents and the radii is .............. $sq. \text{ units}$.

On the circle with center $O$,points $A$ and $B$ are such that $OA = AB$. $A$ point $C$ is located on the tangent at $B$ to the circle such that $A$ and $C$ are on the opposite sides of the line $OB$ and $AB = BC$. The line segment $AC$ intersects the circle again at $F$. Then,the ratio $\angle BOF : \angle BOC$ is equal to

The line $ax + by + c = 0$ is a normal to the circle $x^2 + y^2 = r^2$. The portion of the line $ax + by + c = 0$ intercepted by this circle is of length:

The coordinates of the centre of the smallest circle passing through the origin and having $y=x+1$ as a diameter are

The points $(x_1, y_1), (x_2, y_2), (x_1, y_2),$ and $(x_2, y_1)$ are always:

Let $A(2, 3), B(4, 5)$ be two points and let $C = (x, y)$ be a point such that $(x - 2)(x - 4) + (y - 3)(y - 5) = 0$. If the area of $\Delta ABC = \sqrt{2}$ sq. unit,then the maximum number of positions of $C$ in the $xy$ plane is