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A ray of light passing through the point $\mathrm{P}(2,3)$ reflects on the $x$ -axis at point $\mathrm{A}$ and the reflected ray passes through the point $\mathrm{Q(5,4)}$ . Let $\mathrm{R }$ be the point that divides the line segment $\mathrm{AQ}$ internally into the ratio $\mathrm{2:1}$ . Let the co-ordinates of the foot of the perpendicular $\mathrm{M}$ from $\mathrm{R}$ on the bisector of the angle $\mathrm{P A Q}$ be $\mathrm{(\alpha, \beta)}$ . Then, the value of $\mathrm{7\alpha+3\beta}$ is equal to _____.
Option: 1
31

Option: 2
-

Option: 3
-

Option: 4
-

Answers (1)

best_answer

$\mathrm{{p}'}$ be reflection of p about x-axis

$\mathrm{Slope\: of \; {p}'Q\:\; \; ^{m}{p}'_{Q}= \, ^{m}A_{Q}= \frac{4+3}{5-2}= \frac{7}{3}.}$
$\mathrm{\Rightarrow AQ:y-4= \frac{7}{3}\left ( x-5 \right )\quad y= 0\Rightarrow x= \frac{-12}{7}+5= \frac{23}{7}}$ .

So Coordinates of $\mathrm{M:\left ( \frac{23}{7},\frac{8}{3} \right )= \left ( \alpha ,\beta \right )}$

$\mathrm{\Rightarrow 7\alpha +3\beta = 23+8= 31.}$

Posted by

Ritika Kankaria

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