The distance of the point (1,0,2) from the point of intersection of the line $fraction numerator straight x minus 2 over denominator 3 end fraction space equals space fraction numerator straight y plus 1 over denominator 4 end fraction space equals space fraction numerator straight z minus 2 over denominator 12 end fraction$ and the plane x-y +z = 16 is

$2 square root of 14$

8

$3 square root of 21$

13

Solution

Given equation of line
$fraction numerator straight x minus 2 over denominator 3 end fraction space equals space fraction numerator straight y plus 1 over denominator 4 end fraction space equals space fraction numerator straight z minus 2 over denominator 12 end fraction space equals space straight lambda space.. space left parenthesis straight i right parenthesis$
and the equation of the plane is
x-y+z = 16 ... (ii)
Any point on the line (i) is
(3λ +2, 4λ-1, 12λ +2) = 16
11λ + 5 = 16
11λ =11
λ =1
therefore, Point of intersection is (5,3,14)
Now, distance between the points (1,0,2) and (5,3,2)
$space equals square root of left parenthesis 5 minus 1 right parenthesis squared left parenthesis 3 minus 0 right parenthesis squared plus left parenthesis 14 minus 2 right parenthesis squared end root equals space square root of 16 plus 9 plus 144 end root equals space square root of 169 space equals space 13$

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