Question

If x= a (cos 0 + 0 sin 0), y = a (sin 0 - 0 cos 0 ) then [ d² y/dx²]0=π/4= ?​

Answer 1

Answer:

[ d²y/dx² ] at θ = π/4 = 8√2/aπ

Step-by-step explanation:

x = a(cosθ + θsinθ)

diff wrt θ

dx/dθ = a[ - sinθ + (θcosθ +sinθ))]

= a[- sinθ + +θ cosθ +sinθ]

= aθcosθ

y = a(sinθ - θ cosθ)

diff wrt θ

dy/dθ = a{cosθ - [θ (-sinθ) + cosθ]}

= a{cosθ +θsinθ -cosθ}

= aθsinθ

∴ dy/dx = (dy/dθ)/(dx/dθ)

= aθsinθ/aθcosθ

= tanθ

d²y/dx² = sec²θ x dθ/dx

= sec²θ /aθcosθ

= 1/aθcos³θ

∴ [ d²y/dx² ] at θ = π/4 = 1/(aπ/4 xcos³π/4)

= 4 x 2√2/aπ

= 8√2/aπ