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

  • Option 1)

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

  • Option 2)

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

  • 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 of x=y2

Distance between (a2, a) and y-x-1=0 is

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

This is minima when a=\frac{1}{2}

distance=\frac{3}{4\sqrt{2}}=\frac{3\sqrt{2}}{8}

 


Option 1)

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

This option is incorrect

Option 2)

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

This option is incorrect

Option 3)

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

This option is incorrect

Option 4)

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

This option is correct

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