Question

Polar & rectangular equations

Answer 1

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`

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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)`