यदि `sqrt(1-x^(4))+sqrt(1-y^(4))=a(x^(2)-y^(2))`, तो सिद्ध कीजिये कि `(dy)/(dx)=x/ysqrt((1-y^(4))/(1-x^(4)))`

Explanation

` sqrt(1-x^(4)) + sqrt(1-y^(4))=a(x^(2)- y^(2))" "`…(1)
` x^(2)= sin theta " "` …(2)
और ` y^(2) sin emptyset " " `…(3)
तब समीकरण (1) से ]
` sqrt(1- sin ^(2) theta ) + sqrt(1- sin ^(2)) emptyset = a(sin theta - sin emptyset )`
` rArr cos theta + cos emptyset = a(sin theta - sin emptyset )`
` rArr 2 cos (( theta + emptyset )/2) cos ((theta - emptyset )/2)`
` = 2 a (( 0 + emptyset )/2) sin (( theta - emptyset )/2)`
` rArr cos ((0 - emptyset )/2)= a sin ((theta - emptyset)/2)`
`rArr cot ((theta - emptyset)/2) = a `
` rArr (theta - emptyset)/0 = cot ^(-1) a `
`rArr theta - emptyset = 2 cot^(-1)a`
` rArr sin ^(-1) - sin ^(-1)y^(2) = 2 cot^(-1) a` [समीकरण (2) व (3) से ]
दोनों पक्षों का x के सापेक्ष अवकलन करने पर ,
` 1/ sqrt(1-(x)^(2))^(2)*2x - 1/sqrt(1-(y^(2))^(2))*2y (dy)/(dx)=0`
` rArr x/sqrt(1- x^(4))= y/sqrt(1-(y^(4)))(dy)/(dx)`
`rArr (dy)/(dx)= x/y sqrt((1-y^(4))/(1-x^(4)))`