Question

Find the radius of curvature at x = 1 on the curve x⁴+y ⁴ = xy. ⁴

Answer 1

To find the radius of curvature at x = 1 on the given curve, differentiate the equation with respect to x twice and use the formula r = (1+(dy/dx)²)^(3/2)/|(d²y/dx²)|.

Step-by-step explanation:

To find the radius of curvature at x = 1 on the curve x⁴+y ⁴ = xy ⁴, we need to differentiate the equation twice w.r.t x and then use the formula:

r = (1+(dy/dx)²)^(3/2)/|(d²y/dx²)|

1. Differentiate the equation once: 4x³ + 4y³(dy/dx) = y + xy³(dy/dx)³
2. Differentiate the equation again to find d²y/dx²: 12x² + 12y²(dy/dx)² = -3y² - 3x²y³(dy/dx)² + xy²(dy/dx)² + 3xy³ dy²/dx
3. Substitute x = 1 into the equations obtained and solve for dy/dx and d²y/dx².
4. Substitute the values of dy/dx and d²y/dx² into the formula to find the radius of curvature.