Question

Convert polar equation to the standard form of each rectangular equation: r=2cosθ-3sinθ

Answer 1

`\\ \rm\Rrightarrow r=2cos\theta-3sin\theta`

• a(rcosTheta) +b(r sintheta)=ax+by

`\\ \rm\Rrightarrow (2)/(r)cos\theta-(3)/(r) sin\theta=1`

`\\ \rm\Rrightarrow 2x-3y=1`

`\\ \rm\Rrightarrow 2x-3y-1=0`

Answer 2

`(x-1)^2+(y+1.5y)^2=3.25`

Explanation:

`x=r \cos(\theta)`

`y=r \sin(\theta)`

Given polar equation:

`r=2\cos(\theta)-3 \sin(\theta)`

Multiply both sides by r:

`r^2=2r\cos(\theta)-3r \sin(\theta)`

Know that
`x^2+y^2=r^2`:

`x^2+y^2=2r\cos(\theta)-3r \sin(\theta)`

Substitute
`x=r \cos(\theta)` and
`y=r \sin(\theta)` :

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

Therefore,

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

Complete the square:

`(x-1)^2-1+(y+1.5y)^2-2.25=0`

`(x-1)^2+(y+1.5y)^2=3.25`