Area of the circle in which a chord of length $\sqrt{2}$ makes an angle $\frac{\pi}{2}$ at the centre is

If the radical axis of the circles $x^2 + y^2 - 1 = 0$ and $x^2 + y^2 - 2x - 2y + 1 = 0$ forms a triangle of area $A$ with the coordinate axes,then the value of $\frac{1}{A}$ is

Suppose $ABCDEF$ is a hexagon such that $AB=BC=CD=1$ and $DE=EF=FA=2$. If the vertices $A, B, C, D, E, F$ are concyclic,the radius of the circle passing through them is

Let the line $x - y = 4$ intersect the circle $C : (x - 4)^2 + (y + 3)^2 = 9$ at the points $Q$ and $R$. If $P(\alpha, \beta)$ is a point on $C$ such that $PQ = PR$,then $(6\alpha + 8\beta)^2$ is equal to . . . . . .

From a point $P$ on the line $4x - 3y = 6$,two tangents are drawn to the circle $x^2 + y^2 - 6x - 4y + 4 = 0$. If the angle between these tangents is $\tan^{-1}\left(\frac{24}{7}\right)$,then $P$ can be:

If the diameter of the circle $2(x^2 + y^2) + 3x + 4y - 1 = 0$ is $y = 2x + k$,then $k = \dots$