   Chapter 10.3, Problem 78E

Chapter
Section
Textbook Problem

# (a) Use Exercise 77 to show that the angle between the tangent line and the radial line is ψ = π/4 at every point on the curve r = eθ. (b) Illustrate part (a) by graphing the curve and the tangent lines at the points where θ = 0 and π/2. (c) Prove that any polar curve r = f(θ) with the property that the angle ψ between the radial line and the tangent line is a constant must be of the form r = Cekθ, where C and k are constants.

(a)

To determine

To show: The angle between the tangent line and the radial line is ψ=π4 at every point on the curve r=eθ .

Explanation

Given:

The radial line OP and the angle between the tangent lines P is shown in below figure 1.

The radial line OP and the angle between the tangent lines P is tanψ=rdrdθ .

The radial line is ψ=π4 .

Point on the curve is r=eθ .

Calculation:

The radial line OP and the angle between the tangent lines P is tanψ=rdrdθ .

Point on the curve is r=eθ .

Substitute (drdθ) for (r) and in equation r=eθ

r=eθ(drdθ)=eθ

Substitute

(b)

To determine

To plot: The curve and the tangent line at the point where θ=0 and θ=π2 .

(c)

To determine

To prove: Any polar curve r=f(θ) with the property and angle between radial line and tangent line is a constant.

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