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Application Of Derivatives

Question
CBSEENMA12035521
Wired Faculty App

Find the points on the curve 1 over straight x squared plus 1 over straight y squared space equals 1 comma space which are nearest from the origin. 

Solution

The equation of curve is
       1 over straight x squared plus 1 over straight y squared equals space 1 space space space space space space or space space space 1 over straight y squared space equals space 1 minus 1 over straight x squared space space rightwards double arrow space space space space 1 over straight y squared space equals space fraction numerator straight x squared minus 1 over denominator straight x squared end fraction space space space rightwards double arrow space space space straight y squared space equals space straight x squared over straight x squared minus 1
therefore space space space space straight y space equals space plus-or-minus fraction numerator straight x over denominator square root of straight x squared minus 1 end root end fraction space open vertical bar straight x close vertical bar space greater than space 1                                      ...(1)
Let D be the distance of any point (x, y) on the curve 1 over straight x squared plus 1 over straight y squared space equals space 1 from the origin (0, 0).
therefore space space space space space straight D space equals square root of left parenthesis straight x minus 0 right parenthesis squared space left parenthesis straight y minus 0 right parenthesis squared end root space equals space square root of straight x squared plus straight y squared end root space equals space square root of straight x squared plus fraction numerator straight x squared over denominator straight x squared minus 1 end fraction end root space space space space space space space left square bracket because space of space left parenthesis 1 right parenthesis right square bracket space space space space space space space space space space space space space equals square root of fraction numerator straight x to the power of 4 minus straight x squared plus straight x squared over denominator straight x squared minus 1 end fraction end root space equals space square root of fraction numerator straight x to the power of 4 over denominator straight x squared minus 1 end fraction end root space equals space fraction numerator straight x squared over denominator square root of straight x squared minus 1 end root end fraction dD over dx space equals space fraction numerator square root of straight x squared minus 1 end root. space space 2 straight x minus straight x squared space begin display style fraction numerator 2 straight x over denominator 2 square root of straight x squared minus 1 end root end fraction end style over denominator straight x squared minus 1 end fraction space equals space fraction numerator begin display style fraction numerator 2 straight x square root of straight x squared minus 1 end root over denominator 1 end fraction end style minus begin display style fraction numerator straight x cubed over denominator square root of straight x squared minus 1 end root end fraction end style over denominator straight x squared minus 1 end fraction

          equals space fraction numerator 2 straight x left parenthesis straight x squared minus 1 right parenthesis minus straight x cubed over denominator left parenthesis straight x squared minus 1 right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction space equals space fraction numerator straight x cubed minus 2 straight x over denominator left parenthesis straight x squared minus 1 right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction space equals space fraction numerator straight x left parenthesis straight x squared minus 2 right parenthesis over denominator left parenthesis straight x squared minus 1 right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction
Now, dD over dx space equals 0 space space space space space space space space space space space space rightwards double arrow space space space space fraction numerator straight x left parenthesis straight x squared minus 2 right parenthesis over denominator left parenthesis straight x squared minus 1 right parenthesis to the power of begin display style 3 over 2 end style end exponent end fraction space space space space rightwards double arrow space space space space straight x left parenthesis straight x squared minus 2 right parenthesis space equals space 0 space space space space rightwards double arrow space space space straight x space equals space 0 comma space space plus-or-minus square root of 2
Rejecting straight x equals 0 space as space open vertical bar straight x close vertical bar greater than 1 comma space space space we space get space straight x space equals space plus-or-minus square root of 2
fraction numerator straight d squared straight D over denominator dx squared end fraction space equals space fraction numerator left parenthesis straight x squared minus 1 right parenthesis to the power of begin display style 3 over 2 end style end exponent. space space left parenthesis 3 straight x squared minus 2 right parenthesis thin space minus space left parenthesis straight x cubed minus 2 straight x right parenthesis. space begin display style 3 over 2 end style left parenthesis straight x squared minus 1 right parenthesis to the power of begin display style 1 half end style end exponent.2 straight x over denominator left parenthesis straight x squared minus 1 right parenthesis cubed end fraction
               equals space fraction numerator left parenthesis straight x squared minus 1 right parenthesis thin space left parenthesis 3 straight x squared minus 2 right parenthesis space minus space 3 straight x left parenthesis straight x cubed minus 2 straight x right parenthesis over denominator left parenthesis straight x squared minus 1 right parenthesis to the power of begin display style 5 over 2 end style end exponent end fraction space equals space fraction numerator 3 straight x to the power of 4 minus 5 straight x squared plus 2 minus 3 straight x to the power of 4 plus 6 straight x squared over denominator left parenthesis straight x squared minus 1 right parenthesis to the power of begin display style 5 over 2 end style end exponent end fraction equals space fraction numerator straight x squared plus 2 over denominator left parenthesis straight x squared minus 1 right parenthesis to the power of begin display style 5 over 2 end style end exponent end fraction
When straight x equals plus-or-minus square root of 2. space fraction numerator straight d squared straight D over denominator dx squared end fraction space equals space fraction numerator 2 plus 2 over denominator left parenthesis 2 minus 1 right parenthesis to the power of begin display style 5 over 2 end style end exponent end fraction space equals space 4 greater than 0
therefore space space straight D space is space minimum space when space straight x space equals space plus-or-minus square root of 2 comma space space space straight y space equals space plus-or-minus fraction numerator square root of 2 over denominator square root of 2 minus 1 end fraction space equals plus-or-minus square root of 2 therefore space space space space required space points space are space left parenthesis plus-or-minus square root of 2 comma space space space plus-or-minus square root of 2 right parenthesis.

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