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JEE Main202228 Jun 2022Morning ShiftMathematicsStraight LinesActual

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 A Q 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 P A Q be α , β . Then, the value of 7 α + 3 β is equal to _____.

Correct answer

0

Step-by-step solution

By observation we know the point A α , 0 and β = y -coordinate of R = 2 × 4 + 1 × 0 2 + 1 = 8 3           ⋯ 1 Now P ' is image of P in y = 0 which will be P ' 2 , - 3 ∴ Equation of P ' Q is y + 3 = 4 + 3 5 - 2 x - 2 i.e. 3 y + 9 = 7 x - 14 Now P ' Q meets x - axis at A so A ≡ 23 7 , 0 by solving with y = 0 ∴ α = 23 7           ⋯ 2 By 1 , 2 , we get 7 α + 3 β = 23 + 8 = 31

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