   Chapter 10.3, Problem 65E

Chapter
Section
Textbook Problem

Show that the polar equation r = a sin θ + b cos θ, where ab ≠ 0, represents a circle, and find its center and radius.

To determine

To find: the center and radius for the polar equation r=asinθ+bcosθ .

Explanation

Given:

The polar equation r=asinθ+bcosθ (1)

Calculation:

The Cartesian coordinates for the variable x is as below.

x=rcosθxr=cosθ

The Cartesian equation for the variable y is as below.

y=rsinθyr=sinθ

Substitute (xr) for cosθ and (yr) for sinθ in equation (1).

r=asinθ+bcosθ=a(xr)+b(yr)=axr+byr=ax+byrr2=ax+by

Write the radius formula as below.

r2=x2+y2

Substitute (x2+y2) for r in equation r2=ax+by .

r2=ax+byx2+y2=ax+byx2ax+y2by=0

Add (b24b24) and (a24a24) in above equation

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