Consider a circle $C$ which touches the $y$-axis at $(0,6)$ and cuts off an intercept $6 \sqrt{5}$ on the $x$-axis. Then the radius of the circle $C$ is equal to:

If two parallel chords of a circle,having diameter $4 \, \text{units}$,lie on the opposite sides of the centre and subtend angles $\cos^{-1}\left(\frac{1}{7}\right)$ and $\sec^{-1}(7)$ at the centre respectively,then the distance between these chords is:

If the lengths of the tangents drawn from the point $(1,2)$ to the circles $x^2+y^2+x+y-4=0$ and $3x^2+3y^2-x-y-\lambda=0$ are in the ratio $3:4$,then $\lambda$ is equal to

Let $C$ be the circle $x^2+y^2=1$ in the $XY$-plane. For each $t \geq 0$,let $L_t$ be the line passing through $(0,1)$ and $(t, 0)$. Note that $L_t$ intersects $C$ in two points,one of which is $(0,1)$. Let $Q_t$ be the other point. As $t$ varies between $1$ and $1+\sqrt{2}$,the collection of points $Q_t$ sweeps out an arc on $C$. The angle subtended by this arc at $(0,0)$ is

The circle$(s)$ touching the $x$-axis at a distance of $3$ from the origin and having an intercept of length $2 \sqrt{7}$ on the $y$-axis is (are):

In the figure,$ABCD$ is a unit square. $A$ circle is drawn with centre $O$ on the extended line $CD$ and passing through $A$. If the diagonal $AC$ is tangent to the circle,then the area of the shaded region is