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JEE Main202431 Jan 2024Evening ShiftMathematicsThree Dimensional GeometryActual

A line passes through A 4 , − 6 , − 2 and B 16 , − 2 , 4 . The point P a , b , c where a , b , c are non-negative integers, on the line A B lies at a distance of 21 units, from the point A . The distance between the points P a , b , c and Q 4 , − 12 , 3 is equal to ______.

Correct answer

0

Step-by-step solution

Given, A line passes through A 4 , − 6 , − 2 and B 16 , − 2 , 4 , So, the equation of line will be x − 4 12 = x + 6 4 = z + 2 6 ⇒ x − 4 6 = x + 6 2 = z + 2 3 So, the direction cosines will be 6 6 2 + 2 2 + 3 2 , 2 6 2 + 2 2 + 3 2 , 3 6 2 + 2 2 + 3 2 ≡ 6 7 , 2 7 , 3 7 Now, given point P a , b , c lies on the line A B and at a distance of 21 from point A So, by using the equation of line A B we get, ⇒ a − 4 6 7 = b + 6 2 7 = c + 2 3 7 = 21 ⇒ 21 × 6 7 + 4 , 2 7 × 21 − 6 , 3 7 × 21 − 2 ≡ a , b , c ⇒ 22 , 0 , 7 ≡ a , b

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