Evaluation the radius of curvature

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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 ...