If `x=asec^3thetaa n dy=atan^3theta,fin d(dy)/(dx)a ttheta=pi/3dot`

Options

A. `(2)/(sqrt(3))`
B. `(1)/(2)`
C. `(sqrt(3))/(2)`
D. `(1)/(sqrt(3))`

Answer

C

Explanation

We have , ` x = a sec^(3) theta " and " y = a tan^(3) theta `
On differentiating both functions w.r.t` theta` , we get
` (dx)/(d theta) = 3a sec^(2) theta (d)/(dtheta) = 3a sec^(3) theta tan theta `
and ` (dy)/(d theta ) = 3a tan^(2) theta (d)/(d theta ) ( tan theta ) = 3a tan^(2) theta sec^(2) theta `
` therefore (dy)/(dx) = ((dy)/(d theta))/((dx)/(d theta)) = (3a tan^(2) theta sec^(2))/(3a sec^(3) theta tan theta )= (tan theta )/(sec theta ) = sin theta `
Hence , ` ((dy)/(dx))_("at " theta = (pi)/(3)) = sin""(pi)/(3) = (sqrt(3))/(2)`