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If sqrt(1-x^(2)) + sqrt(1-y^(2))=a(x-y),...

If `sqrt(1-x^(2)) + sqrt(1-y^(2))=a(x-y)`, then prove that `(dy)/(dx) = sqrt((1-y^(2))/(1-x^(2)))`

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`" "sqrt (1-x^(2)) + sqrt(1-y^(2)) = a(x-y)`
Let `x = sinA" ", " " y = sinB`
`" " sqrt(1 - sin^(2)A) + sqrt(1- sin^(2)B) = a ( sin A - sinB)`
`" " cos A+cosB = a(sinA - sinB)`
`rArr 2 cos ((A+ B)/(2)) cos ((A-B)/(2)) = 2a cos((A+B)/(2)) sin ((A-B)/(2))`
`" " rArr cos((A-B)/(2)) = a sin ((A-B)/(2))`
`rArr cot((A-B)/(2))=a`
`" "rArr (A-B)/(2) =cot^(-1) a`
`rArr A-B= 2 cot^(-1)a `
`rArr sin^(-1) x - sin^(-1)y = 2 cot^(-1)a`
differentiating w.r.t x
`rArr (1)/(sqrt(1-x^(2)) - (1)/(sqrt(1-y^(2))))(dy)/(dx) =0`
`rArr (dy)/(dx) = (sqrt(1-y^(2)))/(sqrt(1-x^(2)))`
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If sqrt(1-x^(2))+sqrt(1-y^(2))=a(x-y), prove that (dy)/(dx)=sqrt((1-y^(2))/(1-x^(2)))

If sqrt(1-x^(2))+sqrt(1-y^(2))=a(x-y), prove that (dy)/(dx)=sqrt((1-y^(2))/(1-x^(2)))

Knowledge Check

  • If sqrt(1-x^(2))+sqrt(1-y^(2))=a(x-y) , then (dy)/(dx) equals

    A
    `sqrt((1-x^(2))(1-y^(2)))`
    B
    `sqrt((1-y^(2))/(1-x^(2)))`
    C
    `sqrt((1-x^(2))/(1-y^(2)))`
    D
    none of these

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