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Question
Choose the correct option from the given alternatives :
If x = a(cosθ + θ sinθ), y = a(sinθ – θ cosθ), then `((d^2y)/dx^2)_(θ = pi/4)` = .........
Options
`(8sqrt(2))/(api)`
`-(8sqrt(2))/(api)`
`(api)/(8sqrt(2))`
`(4sqrt(2))/(api)`
Solution
`(8sqrt(2))/(api)`
[Hint : `"dy"/"dx" = (("dy"/"dθ"))/(("dx"/"dθ")`
= `(a(cosθ + θsinθ - cosθ))/(a(-sinθ + θcosθ + sinθ)`
= `(aθsinθ)/(aθcosθ)`
= tanθ
and
`(d^2y)/(dx^2) = "d"/"dx"(tanθ)`
= `"d"/"dθ"(tanθ) xx "dθ"/"dx"`
= `sec^2θ xx (1)/(aθcosθ)`
= `(1)/(aθ).sec^3θ`
∴ `((d^2y)/dx^2)_("at" θ = pi/4) = (1)/(a(pi/4))(sec pi/4)^3`
= `(4)/(api) xx (sqrt(2))^3`
= `(8sqrt(2))/(api)]`.
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