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  • The shortest distance between the line y = x and the curve is

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

Option: 1

\frac{7}{4\sqrt{2}}


Option: 2

\frac{7}{2\sqrt{2}}


Option: 3

\frac{7}{4\sqrt{3}}


Option: 4

\frac{5}{4\sqrt{2}}


Answers (1)

best_answer

Line y = x

Eq. of tangent to y2 = x - 2 

\\\mathrm{y^2=x-2}\\\mathrm{2yy'=1}\\\mathrm{y'=\frac{1}{2y}=slope}\\\mathrm{Tangent\;at\;P\;is\;parallel\;to\;the\;line\;x=y}\\\mathrm{so,slope\;should\;be\;equal}\\\mathrm{y'=1=\frac{1}{2y}\Rightarrow y=\frac{1}{2}}\\\mathrm{put\;the\;value\;of\;y\;in\;the\;curve\;y^2=x-2}\\\mathrm{\left ( \frac{1}{2} \right )^2=x-2\Rightarrow x=\frac{9}{4}}\\\mathrm{so,\;P=\left ( \frac{9}{4},\frac{1}{2} \right )}\\\mathrm{Perpendicular\;distance\;from\;the\;point\;P\;to\;the\;line\;y=x}\\\mathrm{\left | \frac{\left ( \frac{9}{4}-\frac{1}{2} \right )}{\sqrt{1^2+1^2}} \right |=\frac{7}{4\sqrt2}}

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