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Asked by j.nagaraji1983 | 24 Oct, 2023, 10:53: AM

Expert Answer

begin mathsize 14px style x squared plus y squared space equals space t space plus space 1 over t end style

begin mathsize 14px style open parentheses x squared plus y squared close parentheses squared space equals space open parentheses t space plus space 1 over t close parentheses squared space equals space t squared plus 1 over t squared plus 2 end style

Expanding LHS and substituting for ( t² + 1/t² ) = x⁴ + y⁴ , we get

begin mathsize 14px style x to the power of 4 plus y to the power of 4 plus 2 space x squared y squared space equals space x to the power of 4 plus y to the power of 4 to the power of space end exponent plus space 2 end style

From above expression we get

begin mathsize 14px style x squared y to the power of 2 to the power of space end exponent space equals space 1 end style

.................................(1)

By differentiating above expression on both sides, we get

begin mathsize 14px style x squared space 2 space y space fraction numerator d y over denominator d x end fraction space plus space y squared space left parenthesis 2 x right parenthesis space equals space 0 end style

From above expression, we get

begin mathsize 14px style x space fraction numerator d y over denominator d x end fraction space equals space minus y end style

If we mulitply both sides of above expression by x²y , we get

begin mathsize 14px style x cubed y space fraction numerator d y over denominator d x end fraction space equals space minus x squared y squared end style

Usine eqn.(1) , we re write the above expression as

begin mathsize 14px style x cubed y space fraction numerator d y over denominator d x end fraction space equals space minus space 1 end style

Answered by Thiyagarajan K | 24 Oct, 2023, 02:56: PM

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