Let the line $\mathrm{L}_1: x+3=0$ intersect the lines $\mathrm{L}_2: x-y=0$ and $\mathrm{L}_3: 3 x+y=0$ at the points A and B , respectively. Let the bisector of the obtuse angle between the lines $L_2$ and $L_3$ intersect the line $L_1$ at the point $C$. Then $B C^2: A C^2$ is equal to:

Let the vertex A of a triangle ABC be $(1,2)$, and the mid-point of the side AB be $(5,-1)$. If the centroid of this triangle is $(3,4)$ and its circumcenter is $(\alpha, \beta)$, then $21(\alpha+\beta)$ is equal to :

Suppose that two chords, drawn from the point $(1,2)$ on the circle $x^2+y^2+x-3 y=0$ are bisected by the $y$-axis. If the other ends of these chords are R and S , and the mid point of the line segment RS is $(\alpha, \beta)$, then $6(\alpha+\beta)$ is equal to :

A line with direction ratios $1,-1,2$ intersects the lines $\frac{x}{2}=\frac{y}{3}=\frac{z+1}{3}$ and $\frac{x+1}{-1}=\frac{y-2}{1}=\frac{z}{4}$ at the points P and Q , respectively. If the length of the line segment PQ is $\alpha$, then $225 \alpha^2$ is equal to: