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 ration 2 : 1. Let the co-ordinates of the foot of the perpendicular M from R on the bisector of the angle PAQ be (a, b). Then the value of 7a + 3b is equal to………..

$R\left(\frac{10+\alpha }{3},\frac{8}{3}\right)$

$\left|m_{AQ}\right|=\left|m_{AP}\right|$

$\Rightarrow \left|\frac{4}{5-\alpha }\right|=\left|\frac{3}{2-\alpha }\right|$

= 7 not possible $\therefore \alpha =\frac{23}{7}.\therefore 7\alpha +3\beta =23+8=31$

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