Explanation Procedure used: Figure below shows the reflecting surface. Figure-(1) Consider the angle of incidence equal to the angle of reflection to determine the differential equation that describes the shape of the curve. Calculation: The value of angle between P M and O M is, tan ( π − ϕ ) = P M O M − tan ϕ = − y x tan ϕ = y x (1) Sum of internal angles of the triangle P N O is same as an external angle at O as the angle of incidence equal to the angle of reflection. 2 θ = ϕ tan 2 θ = tan ϕ tan 2 θ = y x 2 tan θ 1 − tan 2 θ = y x (2) The value of slope of tangent at P is, y ′ = tan θ Substitute y ′ for tan θ in equation (2). 2 y ′ 1 − y ′ 2 = y x 2 x y ′ = y − y y ′ 2 y y ′ 2 + 2 x y ′ − y = 0 (3) Solve quadratic equation (3)

6th Edition

ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 1.3, Problem 29E

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The differential equation that describes the shape of the curve.

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