# Evaluation the radius of curvature

Curvature (III)

Differential Calculus

Evaluate the radius of curvature at any point (x,y) for the curves :

(a) xy = c2 (b) y = (1/2)a(ex/a + e-x/a)

See attached file for full problem description.

Â© BrainMass Inc. brainmass.com December 15, 2022, 4:25 pm ad1c9bdddfhttps://brainmass.com/math/derivatives/evaluation-radius-curvature-23921

#### Solution Preview

Please see the attached file.

Curvature (III)

Differential Calculus

Written by :- Thokchom Sarojkumar Sinha

Evaluate the radius of curvature at any point (x,y) for the curves :

(a) xy = c2 (b) y = (1/2)a(ex/a + e-x/a)

Solution :- (a) xy = c2

Differentiating the equation with respect to 'x' , we get

y + x dy/dx = 0 --------------------------------------(1)

or, y + xy1 = 0 where y1 = dy/dx

or, xy1 = - y or, y1 = - y/x ------------------------------(1a)

Again differentiating (1) with respect to 'x' , we get

dy/dx + 1.dy/dx + x d2y/dx2 = 0

or, 2dy/dx + xd2y/dx2 = 0 or, 2y1 + xy2 = 0 where y1 = dy/dx , y2 = d2y/dx2

or, 2.(-y/x) + xy2 = 0 by using the relation y1 = - y/x

or, - 2y/x + xy2 = 0 or, xy2 = 2y/x

or, y2 = 2y/x2 ...

#### Solution Summary

This solution is comprised of a detailed explanation for finding the radius of curvature at any point of the curves.

It contains step-by-step explanation for finding the radius of curvature at any point (x,y) for the curves :

(a) xy = c2 (b) y = (1/2)a(ex/a + e-x/a)

Solution contains detailed step-by-step explanation.