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

Answer 1

$\frac {4}{5 - α} = \frac {3}{α - 2}$
$⇒ 4α - 8 = 15 - 3α$

$α = \frac {23}{7}$

$A = ( \frac {23}{7},0)\ Q = (5,4)$

$R = (\frac {10 + \frac {23}{7}}{3} , \frac 83 )$

$= ( \frac {31}{7} , \frac 83 )$
Bisector of angle PAQ is $X = \frac {23}{7}$.
$⇒ M = ( \frac {23}{7} , \frac 83 )$

So, $7α + 3β = 31$