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JEE Main 2026 · 2 Apr · Shift 1, Mathematics question 15

Let a line LL passing through the point (1,1,1)(1,1,1) be perpendicular to both the vectors 2i^+2j^+k^2\hat i+2\hat j+\hat k and i^+2j^+2k^\hat i+2\hat j+2\hat k. If P(a,b,c)P(a,b,c) is the foot of perpendicular from the origin on the line LL, then the value of 34(a+b+c)34(a+b+c) is :

Choose one option
Answer and worked solution

Official answer: C

Direction of L. (2,2,1)×(1,2,2)=(2,−3,2)(2,2,1)\times(1,2,2)=(2,-3,2).

Foot from O. On (1,1,1)+t(2,−3,2)(1,1,1)+t(2,-3,2), take t=−(1,1,1)⋅(2,−3,2)17=−117t=-\dfrac{(1,1,1)\cdot(2,-3,2)}{17}=-\dfrac1{17}:
P=(1517,2017,1517),a+b+c=5017.P=\left(\tfrac{15}{17},\tfrac{20}{17},\tfrac{15}{17}\right),\quad a+b+c=\frac{50}{17}.

Value. 34(a+b+c)=10034(a+b+c)=100.

(C)

Topics:Angles and distancesCross product
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