From the point $(-1,-1)$, two rays are sent making angles of $45^{\circ}$ with the line $x+y=0$. These rays get reflected from the mirror $x+2 y=1$. If the equations of the reflected rays are $\mathrm{a} x+\mathrm{b} y=9$ and $c x+d y=7, a, b, c, d \in \mathbf{Z}$, then the value of $a d+b c$ is $\_\_\_\_$ .

Let $\mathrm{A}, \mathrm{B}$ be points on the two half-lines $x-\sqrt{3}|y|=\alpha, \alpha>0$ at a distance of $\alpha$ from their point of intersection $P$. The line segment $A B$ meets the angle bisector of the given half-lines at the point $Q$. If $P Q=\frac{9}{2}$ and $R$ is the radius of the circumcircle of $\triangle \mathrm{PAB}$, then $\frac{\alpha^2}{R}$ is equal to $\_\_\_\_$

Let the distance between two parallel lines be 5 units and a point $P$ lie between the lines at a unit distance from one of them. An equilateral triangle $P Q R$ is formed such that $Q$ lies on one of the parallel lines, while R lies on the other. Then $(Q R)^2$ is equal to _________.

Let a ray of light passing through the point $$(3,10)$$ reflects on the line $$2 x+y=6$$ and the reflected ray passes through the point $$(7,2)$$. If the equation of the incident ray is $$a x+b y+1=0$$, then $$a^2+b^2+3 a b$$ is equal to _________.