Observe the following statements:
$I$. The circle $x^2+y^2-6x-4y-7=0$ touches the $y$-axis.
$II$. The circle $x^2+y^2+6x+4y-7=0$ touches the $x$-axis.
Which of the following is a correct statement?

$A$ square is inscribed in the circle $x^2 + y^2 - 2x + 4y - 93 = 0$ with its sides parallel to the coordinate axes. The coordinates of its vertices are

If one of the diameters of the circle $x^{2}+y^{2}-2 \sqrt{2} x-6 \sqrt{2} y+14=0$ is a chord of the circle $(x-2 \sqrt{2})^{2}+(y-2 \sqrt{2})^{2}=r^{2}$,then the value of $r^{2}$ is equal to

Let $AB$ be a chord of a circle and $C$ divides $AB$ internally in the ratio $3 : 1$. $A$ line through $C$ cuts the circle at $D$ and $E$ such that the minimum distances of $D$ and $E$ from line $AB$ are $3$ and $2$ respectively. If $r$ is the minimum length of $AB$ such that $\alpha$ is the angle between $AB$ and $DE$ for this $r$,then the value of '$r\alpha$' is

The equation of the circle inscribed in a square formed by the lines $x+y-2=0$,$x+y-6=0$,$x-y+1=0$ and $x-y+5=0$ is

Let the centre of the circle $S=0$ lie on the line $x+y-5=0$ and also lie in the first quadrant. If this circle touches both the lines $x-2=0$ and $y-5=0$,then the area of the circle is