Explanation Given information: The angle between curves C 1 and C 2 at the point of intersection P is defined as the angle 0 ≤ θ ≤ π 2 between the tangent lines at P . Concept Involved: The angle between the tangent line at the intersection point P is given by tan θ = | m 1 − m 2 1 + m 1 m 2 | Here m 1 and m 2 are the slope of the tangent line of each curve respectively. Since, x 2 + y 2 = 1 ⇒ x 2 = 1 − y 2 And x 2 + ( y − 1 ) 2 = 1 ⇒ x 2 = 1 − ( y − 1 ) 2 So, 1 − y 2 = 1 − ( y − 1 ) 2 1 − y 2 = 1 − y 2 − 1 + 2 y 2 y = 1 ⇒ y = 1 2 Then, x 2 = 1 − 1 4 = 3 4 x = ± 3 2 Therefore, point of intersection are ( 3 2 , 1 2 ) and ( − 3 2 , 1 2 ) . Now find the slope of the tangent line of the given curves at these intersection points. For the curve x 2 + y 2 = 1 , 2 x + 2 y d y d x = 0 d y d x = − x y Then, m 1 = ( d y d x ) ( 3 2 , 1 2 ) = − 3 2 1 2 = − 3 For the curve x 2 + ( y − 1 ) 2 = 1 , 2 x + 2 ( y − 1 ) d y d x = 0 d y d x = − x y − 1 Then, m 2 = ( d y d x ) (

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 3.6, Problem 58PS

To determine

To identify: The angle between the circles x2+y2=1 and x2+(y−1)2=1 .

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Chapter 3.6, Problem 57PS

Chapter 3.6, Problem 59PS

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Chapter 3 Solutions

Calculus

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The angle between the circles x 2 + y 2 = 1 and x 2 + ( y − 1 ) 2 = 1 .

Solution Summary: The author defines the angle between curves C 1 and C 2 at the intersection point P as mathrmtantheta =

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To identify: The angle between the circles x2+y2=1 and x2+(y−1)2=1 .

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Chapter 3 Solutions