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)`

Answer 1

`(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)]`.