The length of the diameter of a circle which touches the $x$-axis at the point $(1, 0)$ and passes through the point $(2, -3)$ is:

Two tangents are drawn from the point $P(-1, 1)$ to the circle $x^{2}+y^{2}-2x-6y+6=0$. If these tangents touch the circle at points $A$ and $B$,and if $D$ is a point on the circle such that the lengths of the segments $AB$ and $AD$ are equal,then the area of the triangle $ABD$ is equal to:

Let the line $x+y=1$ meet the circle $x^2+y^2=4$ at the points $A$ and $B$. If the line perpendicular to $AB$ and passing through the midpoint of the chord $AB$ intersects the circle at $C$ and $D$,then the area of the quadrilateral $ADBC$ 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 circle $4x^2+4y^2-12x-12y+9=0$

The chord of the circle $x^{2}+y^{2}-4x=0$ which is bisected at $(1,0)$ is perpendicular to the line