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

  • Option 1)

    \frac{2\sqrt{3}}{8}

  • Option 2)

    \frac{3\sqrt{2}}{5}

  • Option 3)

    \frac{\sqrt{3}}{4}

  • Option 4)

    \frac{3\sqrt{2}}{8}

 

Answers (1)

As we learnt in

Perpendicular distance of a point from a line -

\rho =\frac{\left | ax_{1}+by_{1}+c\right |}{\sqrt{a^{2}+b^{2}}}

 

 

- wherein

\rho  is the distance from the line ax+by+c=0 .

 

 Let (a2 , a) be a point on x = y2

Distance between (a2, a) and x- y+ 1= is

\frac{a^{2}-a+1}{\sqrt{2}}= \frac{1}{\sqrt{2}}[(a-\frac{1}{2})^{2}+\frac{3}{4}]

It is minimum when a=\frac{1}{2}

So minimum distance = \frac{3}{4\sqrt{2}}

 


Option 1)

\frac{2\sqrt{3}}{8}

This is incorrect.

Option 2)

\frac{3\sqrt{2}}{5}

This is incorrect.

Option 3)

\frac{\sqrt{3}}{4}

This is incorrect.

Option 4)

\frac{3\sqrt{2}}{8}

This is incorrect.

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