Question

Find the radius of curvature at any point on the curve X=(acost)/t. Y=(asint)/t​

Answer 1

Step-by-step explanation:

Given,

   x = a (cost + t sint)

   y = a (sint - t cost)

Then, x' (derivative of x w.r. to t)

   = a (- sint + sint + t cost)

   = at cost

and y' (derivative of y w.r. to t)

   = a (cost - cost + t sint)

   = at sint

Also, x" (derivative of x' w.r. to t)

   = a (cost - t sint)

and y" (derivative of y' w.r. to t)

   = a (sint + t cost)

Hence, radius of curvature (ρ)

   =  

   =  

   =  

   = at, at the point 't'.  Given,

   x = a (cost + t sint)

   y = a (sint - t cost)

Then, x' (derivative of x w.r. to t)

   = a (- sint + sint + t cost)

   = at cost

and y' (derivative of y w.r. to t)

   = a (cost - cost + t sint)

   = at sint

Also, x" (derivative of x' w.r. to t)

   = a (cost - t sint)

and y" (derivative of y' w.r. to t)

   = a (sint + t cost)

Hence, radius of curvature (ρ)

   =  

   =  

   =  

   = at, at the point 't'.