The shortest distance between the line y – x = 1and the curve x = y² is

$fraction numerator 3 square root of 2 over denominator 8 end fraction$
$fraction numerator 2 square root of 3 over denominator 8 end fraction$
$fraction numerator 3 square root of 2 over denominator 5 end fraction$
$fraction numerator square root of 3 over denominator 4 end fraction$

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Solution

fraction numerator 3 square root of 2 over denominator 8 end fraction

1 space equals space 2 straight y dy over dx dy over dx space equals space fraction numerator 1 over denominator 2 straight y end fraction space equals 1 rightwards double arrow space straight y space equals space 1 half rightwards double arrow space straight x space equals 1 fourth shortest space distance space equals space open vertical bar fraction numerator begin display style 1 half minus 1 half minus 1 end style over denominator square root of 2 end fraction close vertical bar space equals space fraction numerator 3 over denominator 4 square root of 2 end fraction space equals space fraction numerator 3 square root of 2 over denominator 8 end fraction

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