The maximum area of a rectangle inscribed in the circle $(x+1)^{2}+(y-3)^{2}=64$ is

If $(x, 3)$ and $(3, 5)$ are the extremities of a diameter of a circle with centre at $(2, y)$,then the values of $x$ and $y$ are:

$A$ 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:
$(A) \frac{1}{PS}+\frac{1}{ST}<\frac{2}{\sqrt{QS \times SR}}$
$(B) \frac{1}{PS}+\frac{1}{ST}>\frac{2}{\sqrt{QS \times SR}}$
$(C) \frac{1}{PS}+\frac{1}{ST}<\frac{4}{QR}$
$(D) \frac{1}{PS}+\frac{1}{ST}>\frac{4}{QR}$

If a circle with radius $2.5$ units passes through the points $(2, 3)$ and $(5, 7)$,then its centre is

If the length of the tangent from $(f, g)$ to the circle $x^2+y^2=6$ is twice the length of the tangent from the same point to the circle $x^2+y^2+3x+3y=0$,then $f^2+g^2+4f+4g+2$ is equal to

Which of the following statements are true and which are false? In each case,give a valid reason for your answer.
$q$: The centre of a circle bisects each chord of the circle.