A ray of light passing through the point P 2 , 3 reflects on the X -axis at point A and the reflected ray…

By observation we know the point $A\left(\alpha ,0\right)$ and $\beta =y$-coordinate of $R$$=\frac{2\times 4+1\times 0}{2+1}=\frac{8}{3} \cdots \left(1\right)$

Now $P^{\text{'}}$ is image of $P$ in $y=0$ which will be $P^{\text{'}}\left(2,-3\right)$

$\therefore$  Equation of $P^{\text{'}}Q$ is $\left(y+3\right)=\frac{4+3}{5-2}\left(x-2\right)$ 

i.e. $3y+9=7x-14$

Now $P^{\text{'}}Q$ meets $x-$axis at $A$

so $A\equiv \left(\frac{23}{7},0\right)$ by solving with $y=0$

$\therefore \alpha =\frac{23}{7} \cdots \left(2\right)$

By $\left(1\right),\left(2\right)$, we get

$7\alpha +3\beta =23+8=31$