$A$ square is inscribed in the circle $x^2+y^2-2x+4y-93=0$ with its sides parallel to the coordinate axes. Which of the following can be one of the vertices of the square?

The equation of the chord of the circle $x^2+y^2-4x=0$ whose midpoint is $(1,0)$ is

Two circles $(x + a)^2 + (y + b)^2 = a^2$ and $(x + \alpha)^2 + (y + \beta)^2 = \beta^2$ cut orthogonally if:

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 .............. $sq. \text{ units}$.

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

$A$ straight line through the vertex $P$ of a triangle $PQR$ intersects the side $QR$ at the point $S$ and the circumcircle of the triangle $PQR$ at the point $T$. If $S$ is not the centre of the circumcircle,then:
$(A) \frac{1}{PS}+\frac{1}{ST}<\frac{2}{\sqrt{QS \times SR}}$
$(B) \frac{1}{PS}+\frac{1}{ST}>\frac{2}{\sqrt{QS \times SR}}$
$(C) \frac{1}{PS}+\frac{1}{ST}<\frac{4}{QR}$
$(D) \frac{1}{PS}+\frac{1}{ST}>\frac{4}{QR}$