Question

Find the circle of curvature at (a, a)
(a, a) on x4 + y4 = 2a²xy.​

Answer 1

Step-by-step explanation:

Solution.

Write the derivatives of the quadratic function:

y′=(x2)′=2x;y′′=(2x)′=2.

Then the curvature of the parabola is defined by the following formula:

K=y′′[1+(y′)2]32=2[1+(2x)2]32=2(1+4x2)32.

At the origin (at x=0), the curvature and radius of curvature, respectively, are

K(x=0)=2(1+4⋅02)32=2,R=1K=12.