Question

A ray of light passing through the point $P(2,3)$ reflects on the $x$-axis at point $A$ and the reflected ray passes through the point $Q(5,4)$. Let $R$ be the point that divides the line segment $AQ$ internally into the ratio $2:1$. Let the co-ordinates of the foot of the perpendicular $M$ from $R$ on the bisector of the angle PAQ be $(\alpha ,\beta )$. Then, the value of $7\alpha +3\beta$ is equal to ______

Answer 1

Answer: 31

By observation we see that $A(\alpha ,0)$.
And $\beta =y$-cordinate of $R$
$=\frac{2\times 4+1\times 0}{2+1}=\frac{8}{3}\text{...(1) }$
Now P' is image of $P$ in $y=0$ which will be $P^{\prime }(2,-3)$
Equation of $P^{\prime }Q$ is $(y+3)=\frac{4+3}{5-2}(x-2)$
i.e. $3y+9=7x-14$
$A\equiv \left(\frac{23}{7},0\right)$ by solving with $y=0$
$\therefore \alpha =\frac{23}{7}....$(2)
By (1), (2)
$7\alpha +3\beta =23+8=31$