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Polar & rectangular equations

Polar & rectangular equations-example-1
User Enesa
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1 Answer

12 votes

Answer:

13.
x^2+y^2+2x-3√(x^2+y^2)=0

14.
r=3\cos(\theta)-2\sin(\theta)

Explanation:

Question 13

Conversion from Polar equation to rectangular equation:


x=r \cos(\theta) \implies \cos(\theta)=(x)/(r)


y=r \sin(\theta) \implies \sin(\theta)=(y)/(r)


x^2+y^2=r^2 \implies r=√(x^2+y^2)

Given:


r=3-2\cos(\theta)


\textsf{Substitute }\cos(\theta)=(x)/(r):


\implies r=3-(2x)/(r)

Multiply both sides by r:


\implies r^2=3r-2x


\implies r^2+2x-3r=0


\textsf{Substitute }x^2+y^2=r^2\:\textsf{and }r=√(x^2+y^2):


\implies x^2+y^2+2x-3√(x^2+y^2)=0

---------------------------------------------------------------------------------------------

Question 14

Conversion from Rectangular equation to polar equation:


x=r \cos(\theta)


y=r \sin(\theta)


x^2+y^2=r^2 \implies r^2\cos^2(\theta)+r^2\sin^2(\theta)=r^2

Given:


x^2+y^2-3x+2y=0


\textsf{Substitute }x^2+y^2=r^2\:\textsf{and }x=r \cos(\theta)\:\textsf{and }y=r \sin(\theta):


\implies r^2-3r\cos(\theta)+2r\sin(\theta)=0

Factor out common term r:


\implies r(r-3\cos(\theta)+2\sin(\theta))=0

Divide both sides by r:


\implies r-3\cos(\theta)+2\sin(\theta)=0

Rewrite to make r the subject:


  • \implies r=3\cos(\theta)-2\sin(\theta)
User Bbs
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