Question

यदि (If) `cosx=(1-t^(2))/(1+t^(2))` तथा (and) `siny=(2t)/(1+t^(2)),0 le t le 1` सिद्ध करें कि (Prove that) `(d^(2)y)/(dx^(2))`, t से स्वतंत्र है |

Answer 1

दिया है, `x=cos^(-1)(1-t^(2))/(1+t^(2))`
`:." "x=2 tan^(-1)t" ":." "(dx)/(dt)=(2)/(1+t^(2))" "...(1)`
तथा `y=sin^(-1)(2t)/(1+t^(2))=2 tan^(-1)t`
`:." "(dx)/(dt)=(2)/(1+t^(2))" "...(2)`
अब, `(dy)/(dx)=((dy)/(dt))/((dx)/(dt))=((2)/(1+t^(2)))/((2)/(1+t^(2)))=1`
`:." "(d^(2)y)/(dx^(2))=0`, जो कि t से स्वतंत्र है |
Second method : यहाँ `x=2 tan^(-1)t` तथा `y=2tan^(-1)t`
`:." "y=x" ":." "(dy)/(dx)=1" ""अत: "(d^(2)y)/(dx^(2))=0`