Question

find the radius of curvature y^3=x(x+2y)​

Answer 1

Answer: R=1/4

Step-by-step explanation:  Notice that the graph of the equation

3x2−2x+4y2=0

is an ellipse passing through the origin and tangent to the y-axis therefore dy/dx does not exist. We use implicit differentiation to find x′=dx/dy and x′′=d2x/dy2at the origin. We get

6xx′−2x′+8y=0

and

6(x′)2+6xx′′−2x′′+8=0

Now we evaluate x′ and x′′ at the origin using the above results. We get x′=0 and x′′=4. Therefore the curvature at the origin is

κ=|x′′|(1+x′2)32=4.

Therefore the radius of curvature at the origin is

R=1/4