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QJEEM22Ma2159: JEE Main 2022 Mathematics - Class 11 JEEM22Ma2159 Solution: A ray of light passing through the point P(2, 3) reflects on…

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\alpha , β\beta ). Then, the value of 7 α\alpha + 3 β\beta is equal to ____________.

Step-by-Step Verified Solution

JEE Main 2022 (Online) 28th June Morning Shift Mathematics - Straight Lines and Pair of Straight Lines Question 60 English Explanation

45−α=3α−2⇒4α−8=15−3α{4 \over {5 - \alpha }} = {3 \over {\alpha - 2}} \Rightarrow 4\alpha - 8 = 15 - 3\alpha

α=237\alpha = {{23} \over 7}

A=(237,0) Q=(5,4)A = \left( {{{23} \over 7},0} \right)\,Q = (5,4)

R=(10+2373,83)R = \left( {{{10 + {{23} \over 7}} \over 3},{8 \over 3}} \right)

=(317,83)= \left( {{{31} \over 7},{8 \over 3}} \right)

Bisector of angle PAQ is X=237X = {{23} \over 7}

⇒M=(237,83)\Rightarrow M = \left( {{{23} \over 7},{8 \over 3}} \right)

So, 7α+3β=317\alpha + 3\beta = 31

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