Find the radius of curvature for the curve x3 + y3 = 3axy on the point

1.	Find the radius of curvature for the curve x3 + y3 = 3axy on the point (3a/2, 3a/2).
Answer» According to question x³ + y³ = 3axy Thus at point (\(\frac{3a}{2}\), \(\frac{3a}{2}\)) Putting the coordinates in the equation \(\frac{27a^3}{8}\) + \(\frac{27a^3}{8}\) = \(\frac{54a^3}{4}\) \(\frac{27a^3 + 27a^3}{8}\) = \(\frac{54a^3}{4}\) \(\frac{54a^3}{8}\) = \(\frac{54a^3}{4}\) Thus a = \(\frac{1}{2}\) Thus the coordinates are (\(\frac{1}{2}\), \(\frac{1}{2}\)). Radius of curvatur=-3a/8√2

Find the radius of curvature for the curve x3 + y3 = 3axy on the point (3a/2, 3a/2).

Answer»

According to question

x³ + y³ = 3axy

Thus at point (\(\frac{3a}{2}\), \(\frac{3a}{2}\))

Putting the coordinates in the equation

\(\frac{27a^3}{8}\) + \(\frac{27a^3}{8}\) = \(\frac{54a^3}{4}\)

\(\frac{27a^3 + 27a^3}{8}\) = \(\frac{54a^3}{4}\)

\(\frac{54a^3}{8}\) = \(\frac{54a^3}{4}\)

Thus

a = \(\frac{1}{2}\)

Thus the coordinates are (\(\frac{1}{2}\), \(\frac{1}{2}\)).

Radius of curvatur=-3a/8√2