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Chapter 10: Q 31. (page 683)

Convert each polar equation into a rectangular equation.

r=82-sin(胃)

Short Answer

Expert verified

The rectangular equation is

4x2+3y2-16y=64

Step by step solution

01

Step 1. Given information

The polar equation is

r=82-sin(胃)

02

Step 2. Convert into the rectangular equation

The polar equation is

r=82-sin(胃)

Multiply denominator on both sides

r脳(2-sin(胃))=82-sin(胃)脳(2-sin(胃))r(2-sin(胃))=82r-rsin(胃)=8

Apply the conversion formula

r=x2+y2,y=rsin(胃)

2x2+y2-y=8

Add both sides by y

2x2+y2-y+y=8+y2x2+y2=8+y

Take square on both sides and simplify

(2x2+y2)2=(8+y)2 4(x2+y2)=64+y2+16y4x2+4y2=64+y2+16y4x2+4y2-y2-16y=644x2+3y2-16y=64

So, the rectangular equation is

4x2+3y2-16y=64

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