   Chapter 10.4, Problem 66E

Chapter
Section
Textbook Problem

# Horizontal and Vertical Tangency In Exercises 69 and 70, find the points of horizontal and vertical tangency to the polar curve. r = a sin θ

To determine

To calculate: Points of horizontal and vertical tangency to polar curve r=asinθ.

Explanation

Given: Polar equation is r=asinθ

Calculation:

Given equation is r=asinθ

As we know that x=rcosθ and y=rsinθ.

So equation can be written as

r=asinθ

Use r=xcosθ

xcosθ=asinθ

x=asinθcosθ

Differentiate the above equation with respect to θ

dxdθ=a(cos2θsin2θ)=acos2θ

Put dxdθ=0 for vertical tangent points. It follows that

acos2θ=0

θ=(π4,7π4).

Substitute θ=(π4,7π4) in the equation, we get

r=asin(π4)=a2

r=asin(7π4)=a2

Points of vertical tangency are (a2,π4),(a2,7π4)

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