Question

यदि (If) `sqrt(1-x^(6))+sqrt(1-y^(6))=a(x^(3)-y^(3))`, सिद्ध करें कि (prove that)
`(dy)/(dx)=(x^(2))/(y^(2))sqrt((1-y^(6))/(1-x^(6)))`

Answer 1

दिया है, `sqrt(1-x^(6))+sqrt(1-y^(6))=a(x^(3)-y^(3))` ... (1)
दोनों तरफ x के सापेक्ष अवकलित करने पर हमें मिलता है,
`(d)/(dx)sqrt(1-x^(6))+(d)/(dx)sqrt(1-y^(6))=a[(d)/(dx)(x^(3))-(d)/(dx)(y^(3))]`
या `(d)/(d(1-x^(6)))sqrt(1-x^(6))*(d)/(dx)(1-x^(6))+(d)/(d(1-y^(6)))sqrt(1-y^(6))*(d)/(dy)(1-y^(6))*(dy)/(dx)=a[3x^(2)-(d)/(dy)(y^(3))*(dy)/(dx)]`
या `(1)/(2sqrt(1-x^(6)))*(-6x^(5))+(1)/(2sqrt(1-y^(6)))*(-6y^(5))(dy)/(dx)=a*3x^(2)-3y^(2)*a(dy)/(dx)` या `(3ay^(2)-(3y^(5))/(sqrt(1-y^(6))))(dy)/(dx)=3ax^(2)+(3x^(5))/(sqrt(1-x^(6)))`
या `[3y^(2)((sqrt(1-x^(6))+sqrt(1-y^(6)))/(x^(3)-y^(3)))-(3y^(5))/(sqrt(1-y^(6)))](dy)/(dx)`
`=3x^(2)((sqrt(1-x^(6))+sqrt(1-y^(6)))/(x^(3)-y^(3)))+(3x^(5))/(sqrt(1-x^(6)))`
[(1) से A का मान रखने पर क्योंकि जो संबंध लाना है उसमें A नहीं है ।]
या `(3y^(2)(sqrt(1-x^(6))sqrt(1-y^(6)))+3y^(2)(1-y^(6))-3y^(5)x^(3)+3y^(8))/((x^(3)-y^(3))sqrt(1-y^(6)))(dy)/(dx)`
या `=(3x^(2)(1-x^(6))+3x^(2)sqrt(1-x^(6))sqrt(1-y^(6))+3x^(8)-3x^(5)y^(3))/((x^(3)-y^(3))sqrt(1-x^(6)))`
`(3y^(2)(sqrt(1-x^(6))sqrt(1-y^(6))+1-3xy^(3)))/(sqrt(1-y^(6)))(dy)/(dx)`
`=(3x^(2)(1+sqrt(1-x^(6))sqrt(1-y^(6))-3x^(3)y^(3)))/(sqrt(1-x^(6)))`
`therefore(dy)/(dx)=(x^(2))/(y^(2))*sqrt((1-y^(6))/(1-x^(6)))`
Second method :
दिया है, `sqrt(1-x^(6))+sqrt(1-y^(6))=a(x^(3)-y^(3))`
(1) में, `x^(3)=sinA` तथा `y^(3)=sinB` रखने पर हमें मिलता है,
`sqrt(1-sin^(2))A+sqrt(1-sin^(2)B)=a(sinA-sinB)`
या `cosA+cosB=a(sinA-sinB)`
या `2cos((A+B)/(2))cos((A-B)/(2))=2asin((A-B)/(2))cos((A+B)/(2))`
या `cot((A-B)/(2))=a` या `(A-B)/(2)=cot^(-1)(a)`
या `A-B=2cot^(-1)(a)` या `sin^(-1)x^(3)-sin^(-1)y^(3)=2cot^(-1)(a)`
दोनों तरफ x के सापेक्ष अवकलित करने पर हमें मिलता है ,
`(1)/(sqrt(1-x^(6)))*(d)/(dx)(x^(3))-(1)/(sqrt(1-y^(6)))*(d)/(dx)(y^(3))=0`
या `(1)/(sqrt(1-x^(6)))*3x^(2)-(1)/(sqrt(1-y^(6)))*3y^(2)(dy)/(dx)=0`
`(dy)/(dx)=(x^(2))/(y^(2))*sqrt((1-y^(6))/(1-x^(6)))`