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Continuity And Differentiability

Question

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$If space space ax squared plus 2 hxy plus by squared space equals space 1 comma space verify space that space dy over dx cross times dx over dy equals 1$

Solution

We space have space ax squared plus 2 hxy plus by squared equals 1... left parenthesis 1 right parenthesis Differentiating space both space sides space of space left parenthesis 1 right parenthesis comma space straight w. straight r. straight t. straight x space regarding space straight y space as space straight a space function space of space straight x. space we space get comma 2 ax plus 2 straight h open parentheses straight x. dy over dx plus straight y.1 close parentheses plus straight b.2 straight y dy over dx equals 0 or space ax plus hx dy over dx plus hy plus by dy over dx equals 0 or space dy over dx equals negative fraction numerator ax plus hy over denominator hx plus by end fraction 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 space

Again Differentiating both sides of (1), w.r.t. y regarding x as a function of y, we get,

space space space space space 2 straight a. straight x dx over dy plus 2 straight h open parentheses straight x.1 plus straight y dx over dy close parentheses plus 2 by equals 0 or space ax plus dx over dy plus hx plus hy dx over dy plus by equals 0 therefore space dx over dy left square bracket ax plus hy right square bracket equals negative left parenthesis hx plus by right parenthesis space or space dy over dx equals negative fraction numerator hx plus bx over denominator ax plus hy end fraction 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 Multiplying space left parenthesis 2 right parenthesis space and space left parenthesis 3 right parenthesis comma space we space get comma dy over dx cross times dx over dy equals negative fraction numerator ax plus hy over denominator hx plus hy end fraction cross times negative fraction numerator hx plus bx over denominator ax plus hy end fraction equals 1

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