For the circle $x^2+y^2-9=0$,find the equation of the chord having $(1,2)$ as its mid-point.

Let $ABCD$ be a square of side length $1$,and $\Gamma$ be a circle passing through $B$ and $C$,and touching $AD$. The radius of $\Gamma$ is

If $A(-2, 1)$,$B(0, -2)$,and $C(1, 2)$ are the vertices of a triangle $ABC$,then the perpendicular distance from its circumcentre to the side $BC$ is

The equation of the circle which passes through $(1, 0)$ and $(0, 1)$ and has its radius as small as possible,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

Consider the region $R = \{( x , y ) \in \mathbb{R} \times \mathbb{R} : x \geq 0 \text{ and } y^2 \leq 4- x \}$. Let $F$ be the family of all circles that are contained in $R$ and have centers on the $x$-axis. Let $C$ be the circle that has the largest radius among the circles in $F$. Let $(\alpha, \beta)$ be a point where the circle $C$ meets the curve $y^2=4- x$.
$(1)$ The radius of the circle $C$ is. . . . . .
$(2)$ The value of $\alpha$ is. . . . .
Given the answer for $(1)$ and $(2)$: