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Sponsor Area

Continuity And Differentiability

Question
CBSEENMA12035470

If space left parenthesis straight x minus straight a right parenthesis squared space plus left parenthesis straight y minus straight b right parenthesis squared space equals straight c squared space comma space prove space that space space fraction numerator open square brackets 1 plus open parentheses begin display style dy over dx end style close parentheses squared close square brackets to the power of begin display style 3 over 2 end style end exponent over denominator begin display style fraction numerator straight d squared straight y over denominator dx squared end fraction end style end fraction space is space straight a space constant space independent space of space straight a space and space straight b.

Solution
The space given space equation space is space left parenthesis straight x minus straight a right parenthesis squared space plus left parenthesis straight y minus straight b right parenthesis squared space equals straight c squared space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis
Differentiating space it space straight w. straight r. straight t. straight x space successively space two space times comma
2 left parenthesis straight x minus straight a right parenthesis plus 2 left parenthesis straight y minus straight b right parenthesis dy over dx equals 0
or space left parenthesis straight x minus straight a right parenthesis plus left parenthesis straight y minus straight b right parenthesis dy over dx equals 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis
rightwards double arrow space 1 plus left parenthesis straight y minus straight b right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction plus open parentheses dy over dx close parentheses squared equals 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 3 right parenthesis
From space left parenthesis 3 right parenthesis comma space left parenthesis straight y minus straight b right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative open square brackets 1 plus open parentheses dy over dx close parentheses squared close square brackets comma space space therefore space straight y minus straight b equals negative fraction numerator open square brackets 1 plus open parentheses begin display style dy over dx end style close parentheses squared close square brackets to the power of begin display style 3 over 2 end style end exponent over denominator begin display style fraction numerator straight d squared straight y over denominator dx squared end fraction end style end fraction space space space space space space... left parenthesis 4 right parenthesis
Putting space this space value space of space straight y minus straight b space in space left parenthesis 2 right parenthesis comma space we space get comma
left parenthesis straight x minus straight a right parenthesis minus fraction numerator open square brackets 1 plus open parentheses begin display style dy over dx end style close parentheses squared close square brackets to the power of begin display style 2 end style end exponent over denominator begin display style fraction numerator straight d squared straight y over denominator dx squared end fraction end style end fraction. dy over dx equals 0 comma space space therefore space straight x minus straight a equals fraction numerator 1 plus open parentheses begin display style dy over dx end style close parentheses squared over denominator begin display style fraction numerator straight d squared straight y over denominator dx squared end fraction end style end fraction. dy over dx space space space space space space space space space space space space space space space space space space space space space... left parenthesis 5 right parenthesis
Putting space values space of space straight x minus straight a comma space straight y minus straight b space from space left parenthesis 5 right parenthesis space and space left parenthesis 4 right parenthesis space in space left parenthesis 1 right parenthesis comma space we space get.
fraction numerator open square brackets 1 plus open parentheses begin display style dy over dx end style close parentheses squared close square brackets to the power of begin display style 2 end style end exponent over denominator begin display style fraction numerator straight d squared straight y over denominator dx squared end fraction end style end fraction. open parentheses dy over dx close parentheses squared plus fraction numerator open square brackets 1 plus open parentheses begin display style dy over dx end style close parentheses squared close square brackets to the power of begin display style 2 end style end exponent over denominator begin display style open parentheses fraction numerator begin display style straight d squared straight y end style over denominator begin display style dx squared end style end fraction close parentheses squared end style end fraction equals straight c squared
or space space space open square brackets 1 plus open parentheses dy over dx close parentheses squared close square brackets squared open parentheses dy over dx close parentheses squared plus open square brackets 1 plus open parentheses dy over dx close parentheses squared close square brackets squared equals straight c squared open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses squared
or space space space open square brackets 1 plus open parentheses dy over dx close parentheses squared close square brackets squared open square brackets open parentheses dy over dx close parentheses squared plus 1 close square brackets equals straight c squared open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses
or space space space open square brackets 1 plus open parentheses dy over dx close parentheses squared close square brackets cubed equals straight c squared open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses squared
therefore space space open square brackets 1 plus open parentheses dy over dx close parentheses squared close square brackets to the power of 3 over 2 end exponent equals straight c fraction numerator straight d squared straight y over denominator dx squared end fraction space comma space space therefore space fraction numerator open square brackets 1 plus open parentheses begin display style dy over dx end style close parentheses squared close square brackets to the power of begin display style 3 over 2 end style end exponent over denominator begin display style fraction numerator straight d squared straight y over denominator dx squared end fraction end style end fraction equals straight c
which space is space straight a space constant space independent space of space straight a space and space straight b.

Some More Questions From Continuity and Differentiability Chapter