JEE Main 2022 · 28 Jun · Shift 1 · Mathematics Q25

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MathematicsInteger+4 / -1Question 25 of 90

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 ______.

Your answer

Answer and worked solution

Official answer: 31

Point A. The reflected ray lies on the line through $P'(2,-3)$ and $Q(5,4)$ , which meets $y=0$ at $A=\left(\frac{23}7,0\right)$ .

R. $\frac{A+2Q}3=\left(\frac{31}7,\frac83\right)$ .

Bisector of $\angle PAQ$ is the normal to the mirror, $x=\frac{23}7$ ; the foot from R is $\left(\frac{23}7,\frac83\right)$ .

Value. $23+8=31$ .

Answer: 31

Topics:Distance and section formulae Equations of a line

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JEE Main 2022 · 28 Jun · Shift 1, Mathematics question 25