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eliminate the parameter t x=sex^2 t y=tan^2 t

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1 Answer

1) Since sec^2 t = tan^2 t + 1, eliminating t gives x^2 = y^2 + 1.

Answer: x^2 - y^2 = 1 (hyperbola)

 

2) dL = sqrt[(dr)^2 + (rdθ)^2] = sqrt(r'^2 + r^2) dθ, where

r = a(1-cosθ)

r' = asinθ

By symmetry,

L = 2∫[0, pi] sqrt(r'^2 + r^2) dθ

= 2a∫[0, pi] sqrt(sin^2θ + (1-cosθ)^2) dθ

= 2a∫[0, pi] sqrt(2 - 2cosθ) dθ

= 4a∫[0, pi]sqrt((1/2)(1-cosθ)) dθ

= 4a∫[0, pi]sin(θ/2) dθ

= 8a