यदि (If)`x=a(theta-sin theta),y=a(1-cos theta)` तो `(dy)/(dx)` निकालें साथ ही `(d^(2)y)/(dx^(2))` भी निकालें |

Options

A. `-(1)/(4a)"cosec"^(4)(theta)/(2)`
B. `-(1)/(2a)"cosec"^(4)(theta)/(2)`
C. `-(1)/(4a)"cosec"^(4)(theta)`
D. `-"cosec"^(4)(theta)/(2)`

Answer

A

Explanation

`(dx)/(d theta)=a(1-cos theta)" "...(1)`
और `" "(dy)/(d theta)=a sin theta" "...(2)`
अब `(dy)/(dx)=(dy//d theta)/(dx//d theta)=(a sin theta)/(a(1-cos theta))=(2 sin""(theta)/(2)cos""(theta)/(2))/(2 sin^(2)""(theta)/(2))=cot""(theta)/(2)" "...(3)`
दोनों तरफ x के सापेक्ष अवकलित (differentiate) करने पर हमें मिलता है,
`(d^(2)y)/(dx^(2))=(d)/(dx)(cot""(theta)/(2))=(d)/(d)((theta)/(2))(cot""(theta)/(2)).(d((theta)/(2)))/(dx)`
`=-"cosec"^(2)(theta)/(2).(1)/(2).(d theta)/(dx)`
`=-(1)/(2)"cosec"^(2)(theta)/(2).(1)/(a(1-costheta))" "[(1)" से"]`
`=-(1)/(2)"cosec"^(2)(theta)/(2).(1)/(2asin^(2)""(theta)/(2))=-(1)/(4a)"cosec"^(4)(theta)/(2)`