JEE MAIN - Mathematics (2022 - 28th June Morning Shift - No. 16)
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 ($$\alpha$$, $$\beta$$). Then, the value of 7$$\alpha$$ + 3$$\beta$$ is equal to ____________.
Answer
31
Explanation
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$${4 \over {5 - \alpha }} = {3 \over {\alpha - 2}} \Rightarrow 4\alpha - 8 = 15 - 3\alpha $$
$$\alpha = {{23} \over 7}$$
$$A = \left( {{{23} \over 7},0} \right)\,Q = (5,4)$$
$$R = \left( {{{10 + {{23} \over 7}} \over 3},{8 \over 3}} \right)$$
$$ = \left( {{{31} \over 7},{8 \over 3}} \right)$$
Bisector of angle PAQ is $$X = {{23} \over 7}$$
$$ \Rightarrow M = \left( {{{23} \over 7},{8 \over 3}} \right)$$
So, $$7\alpha + 3\beta = 31$$
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