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  4. 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 (α, β). Then, the value of 7 α+3 β is equal to

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Q. 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 ______

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Solution:

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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. $3 y+9=7 x-14$
$A \equiv\left(\frac{23}{7}, 0\right)$ by solving with $y =0$
$\therefore \alpha=\frac{23}{7} ....$(2)
By (1), (2)
$7 \alpha+3 \beta=23+8=31$