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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answer is 31.

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Detailed Solution

P(2,3), Q(5,4) $A (α, 0)$ $P'$ =Image of P with respect to the line y=0 $P' (2, - 3)$

$∴ Equation of P^{'} Q is (y + 3) = \frac{4 + 3}{5 - 2} (x - 2)$ $\Rightarrow 3 y + 9 = 7 x - 14$

Solving $y = 0 a n d 3 y = 7 x - 23$

$A \equiv (\frac{23}{7}, 0) \Rightarrow α = \frac{23}{7}, β = y -cordinate of R$

$β = = \frac{2 \times 4 + 1 \times 0}{2 + 1} = \frac{8}{3}$

$7 α + 3 β = 23 + 8 = 31$

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