   Chapter 12.5, Problem 69E

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# Center of Curvature Let C be a curve given by y = f ( x ) .Let K be the curvature ( K ≠ 0 ) at the point P ( x 0 , y 0 ) and let z = 1 + f ' ( x 0 ) 2 f " ( x 0 ) Show that the coordinates ( α , β ) of the center of curvature at P are ( ( α , β ) = ( x 0 − f ' ( x 0 ) z 0 , y 0 = z ) .

To determine

To prove: The co-ordinates (α,β) of the center of curvature at P are (α,β)=(x0f(x0)z,y0+z ).

Explanation

Given:

The curve C is given by y=f(x), K is the curvature at point P(x0,y0) and z=1+f(x0)2f(x0).

Proof:

P(x0,y0) Is the point on curve y=f(x).

Let (α,β) be the center of curvature.

The radius of curvature is 1K.

Since,

y=f(x)

So, slope of normal line at (x0,y0) is 1f(x0).

Equation of normal line,

yy0 = 1f(x0)(xx0)

Since, (α,β) is on normal line.

Therefore,

f(x0)(βy0)=αx0 ...... (1)

And, (x0,y0) lies on the circle.

Therefore,

(x0α)2+(y0β)2=(1k)2=[(1+f(x0)2)32|f(x0)|]2 .....

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