If the parametric values of two points $A$ and $B$ on the circle $x^2+y^2-6x+4y-12=0$ are $30^{\circ}$ and $90^{\circ}$ respectively,then the equation of chord $AB$ 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:
$(A) \frac{1}{PS}+\frac{1}{ST}<\frac{2}{\sqrt{QS \times SR}}$
$(B) \frac{1}{PS}+\frac{1}{ST}>\frac{2}{\sqrt{QS \times SR}}$
$(C) \frac{1}{PS}+\frac{1}{ST}<\frac{4}{QR}$
$(D) \frac{1}{PS}+\frac{1}{ST}>\frac{4}{QR}$

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 $(2)$:

Let $S=0$ be the circle passing through the points $(2,0)$,$(1,-2)$,and $(-1,1)$. Then the point $(1,2)$

If the length of the chord of the circle $x^{2}+y^{2}=r^{2}$ $(r>0)$ along the line $y-2x=3$ is $r$,then $r^{2}$ is equal to

When is the angle of intersection of two circles $0^{\circ}$?