Question

Find the centre of curvature of y= x^2 of the origin​

Answer 1

Answer:

the curvature will x +8

because of the sentral part is x=eu

Answer 2

Answer:

Centre of curvature is  $\frac{1}{8\sqrt{2} }$

Step-by-step explanation:

Explanation:

Given, $y = x^{2}$ of the origin which means that we need to find the centre of curvature at point (0,0).

We know that ,

Centre of curvature (ρ) = $\frac{[(1+\frac{dy}{dx} )^{2} ]^{\frac{3}{2} } }{\frac{d^{2}y }{dx^{2} } }$  .............(ii)

Step 1 :

$y = x^{2}$ ...........(i)

differentiate  y with respect to x

∴ $\frac{dy}{dx} = 2x$  

again differentiate with respect to x

∴$(\frac{d^{2} y}{dx^{2} } )$ = 2

Step 2:

put the value of  $(\frac{d^{2} y}{dx^{2} } )$  and $\frac{dy}{dx}$  in equation (ii)

Therefore ,

                         (ρ)  =  $\frac{[(1+\frac{dy}{dx} )^{2} ]^{\frac{3}{2} } }{\frac{d^{2}y }{dx^{2} } }$

              ⇒         (ρ)  =    $\frac{[1+(2x)^{2} ]^{\frac{3}{2} } }{2}$

The centre of curvature at x = 0

   =        $\frac{[1+(2 . 0)^{2} ]^{\frac{3}{2} } }{2}$              

 =$\frac{1}{(2)^{\frac{3}{2} } }$ = $\frac{1}{8\sqrt{2} }$

Final answer :

Hence , the centre of curvature of $y = x^{2}$ of the origin is  $\frac{1}{8\sqrt{2} }$.