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Sponsor Area

Continuity And Differentiability

Question
CBSEENMA12035319

Expressing space the space equation space in space terms space of space an space equation space square root of 1 minus straight x squared end root plus square root of 1 minus straight y squared end root equals straight a left parenthesis straight x minus straight y right parenthesis space in space term space oa space an space equation
involving space inverse space trigonometric space functions space by space straight a space suitable space substitution comma space prove space that
dy over dx equals fraction numerator square root of 1 minus straight y squared end root over denominator square root of 1 minus straight x squared end root end fraction.

Solution
Here space square root of 1 minus straight x squared end root plus square root of 1 minus straight y squared end root equals straight a left parenthesis straight x minus straight y right parenthesis
Put space straight x equals sin space straight theta comma space straight y equals sin space straight ϕ
therefore space space space square root of 1 minus sin squared space straight theta end root plus square root of 1 minus sin squared space straight theta end root equals straight a left parenthesis sin space straight theta minus sin space straight ϕ right parenthesis
therefore space space space cos space straight theta plus cos space straight ϕ equals straight a left parenthesis sin space straight theta minus sin space straight ϕ right parenthesis
therefore space space space 2 space cos space fraction numerator straight theta plus straight ϕ over denominator 2 end fraction cos fraction numerator straight theta minus straight ϕ over denominator 2 end fraction equals straight a. space 2 space cos space fraction numerator straight theta plus straight ϕ over denominator 2 end fraction sin fraction numerator straight theta minus straight ϕ over denominator 2 end fraction
or space space space cos fraction numerator straight theta minus straight ϕ over denominator 2 end fraction equals straight a space sin fraction numerator straight theta minus straight ϕ over denominator 2 end fraction space space space or space fraction numerator cos fraction numerator straight theta minus straight ϕ over denominator 2 end fraction over denominator sin fraction numerator straight theta minus straight ϕ over denominator 2 end fraction end fraction equals straight a space space space rightwards double arrow space cot fraction numerator straight theta minus straight ϕ over denominator 2 end fraction equals straight a
rightwards double arrow space space space fraction numerator straight theta minus straight ϕ over denominator 2 end fraction equals cot to the power of negative 1 end exponent straight a space space space rightwards double arrow space straight theta minus straight ϕ equals 2 space cot to the power of negative 1 end exponent straight a space rightwards double arrow space sin to the power of negative 1 end exponent straight x minus sin to the power of negative 1 end exponent straight y equals 2 space cot to the power of negative 1 end exponent straight a.
Diff. space both space sides space straight w. straight r. straight t. straight x. comma space fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction minus fraction numerator 1 over denominator square root of 1 minus straight y squared end root end fraction dy over dx equals 0
therefore space fraction numerator 1 over denominator square root of 1 minus straight y squared end root end fraction dy over dx equals fraction numerator 1 over denominator square root of 1 minus straight x squared end root end fraction space space space space space space space space space space space space space space space space space space therefore space space space space space space space space space dy over dx equals fraction numerator square root of 1 minus straight y squared end root over denominator square root of 1 minus straight x squared end root end fraction

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