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The shortest distance between the line $x - y = 1$ and the curve $x^{2} = 2 y$ is _____

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$\frac{1}{2}$

$\frac{1}{2 \sqrt{2}}$

$\frac{1}{\sqrt{2}}$

answer is B.

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Detailed Solution

$x^{2} = 2 y$

differentiate with respect to x

$2 x = 2 \frac{d y}{d x} \Rightarrow \frac{d y}{d x} = x = 1$ , $∴ y = \frac{1}{2}$

$S . D = \frac{|1 - \frac{1}{2} - 1|}{\sqrt{2}} = \frac{1}{2 \sqrt{2}}$

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