The shortest distance between line `y-x=1` and curve `x=y^2` is (a) `(3sqrt2)/8` (b) `8/(3sqrt2)` (c) `4/sqrt3` (d) `sqrt3/4`

Explanation

General point on curve is `(alpha^2,alpha)`,
distance of it from line`y-x-1=0` is
`d=abs((alpha-alpha^2-1)/sqrt2)=abs((alpha^2-alpha+1)/sqrt2)`
`d'=(2alpha-1)/sqrt2` , for minimum it is `0`
`=>alpha=1/2`
`=>d=abs((1/4-1/2+1)/sqrt2)=abs((1-1/4)/sqrt2)`
`=3/(4sqrt2)`
`=>d=(3sqrt2)/8`