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JEE Main 2022 · 28 Jun · Shift 2, Mathematics question 21

Let the image of the point P(1,2,3)P(1,2,3) in the line L:x−63=y−12=z−23L:\dfrac{x-6}3=\dfrac{y-1}2=\dfrac{z-2}3 be Q. Let R(α,β,γ)R(\alpha,\beta,\gamma) be a point that divides internally the line segment PQ in the ratio 1:31:3. Then the value of 22(α+β+γ)22(\alpha+\beta+\gamma) is equal to ______.

Answer and worked solution

Official answer: 125

Foot of the perpendicular. On (6+3t,1+2t,2+3t)(6+3t,1+2t,2+3t), (5+3t,2t−1,3t−1)⋅(3,2,3)=10+22t=0(5+3t,2t-1,3t-1)\cdot(3,2,3)=10+22t=0, t=−511t=-\frac5{11}: M=(5111,111,711)M=\left(\frac{51}{11},\frac1{11},\frac7{11}\right).

R divides PQ in 1:31:3, so it is the midpoint of PM: (3111,2322,2011)\left(\frac{31}{11},\frac{23}{22},\frac{20}{11}\right), sum 12522\frac{125}{22}.

Answer: 125

Topics:Angles and distancesEquation of a line in space
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