To determine To sketch: The curve x 2 + ( y − c ) 2 = c 2 and its orthogonal trajectory. Explanation The given equation is, x 2 + ( y − c ) 2 = c 2 . Rewrite the function as, x 2 + y 2 − 2 y c = 0 . The orthogonal trajectory of a curve y = f ( x ) is given by y ′ = − 1 f ′ ( x ) . The derivative of x 2 + y 2 − 2 y c = 0 is, x 2 + y 2 − 2 y c = 0 2 x + 2 y d y d x − 2 c d y d x = 0 x + 2 d y d x ( y − c ) = 0 d y d x = x c − y From x 2 + y 2 − 2 y c = 0 , obtain the value of c , x 2 + y 2 2 y = c . Substitute x 2 + y 2 2 y = c in d y d x = x c − y , d y d x = x x 2 + y 2 2 y − y = x x 2 + y 2 − 2 y 2 2 y = 2 x y x 2 + y 2 − 2 y 2 = 2 x y x 2 − y 2 The orthogonal trajectory of d y d x = 2 x y x 2 − y 2 is, d y d x = y 2 − x 2 2 x y . Let v = y x and rewrite the equation d y d x = y 2 − x 2 2 x y as, d y d x = y 2 x 2 − 1 2 x y x 2 d y d x = y 2 x 2 − 1 2 y x d y d x = v 2 − 1 2 v Differentiating v = y x , v = y x d y d x = v + x d v d x This implies, v + x d v d x = v 2 − 1 2 v

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ISBN: 9781119096023

Author: Kreyszig

Publisher: WILEY

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Chapter 1.6, Problem 10P

To determine

To sketch: The curve x2+(y−c)2=c2 and its orthogonal trajectory.

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Chapter 1.6, Problem 9P

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