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JEE Main202330 Jan 2023Evening ShiftMathematicsThree Dimensional GeometryActual

Let a line L pass through the point P ( 2 , 3 , 1 ) and be parallel to the line x + 3 y - 2 z - 2 = 0 = x - y + 2 z . If the distance of L from the point ( 5 , 3 , 8 ) is α , then 3 α 2 is equal to ________

Correct answer

0

Step-by-step solution

Given, x + 3 y - 2 z - 2 = 0 = x - y + 2 z Now direction ratio is given by, i ^ j ^ k ^ 1 3 - 2 1 - 1 2 = 4 i ^ - 4 j ^ - 4 k ^ So, Equation of line will be x - 2 4 = y - 3 - 4 = z - 1 - 4 ⇒ x - 2 1 = y - 3 - 1 = z - 1 - 1 = k Let Q be ( 5 , 3 , 8 ) and foot of ⊥ from Q on this line be R Now, R ≡ ( k + 2 , - k + 3 , - k + 1 ) So, direction ratio of Q R will be ( k - 3 , - k , - k - 7 ), Now using the perpendicular condition we get, ( 1 ) ( k - 3 ) + ( - 1 ) ( - k ) + ( - 1 ) ( - k - 7 ) = 0 &#8658

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