Question

1)a. Find a Cartesian equation for r=3sin(theta)+2cos(theta)? b. complete the square and describe the shape in detail?

Answer 1

Circle

Explanation:

Given polar equation:

`r=3\sin\theta+2\cos\theta`

`r\cdot r=r(3\sin\theta+2\cos\theta)`

`r^2=3r\sin\theta+2r\cos\theta`

Setting `r\cos\theta=x` & `r\sin\theta=y` & `r^2=x^2+y^2` in the above equation, we get the Cartesian form of equation as follows

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

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

`x^2-2x+1-1+y^2-3y+(3/2)^2-(3/2)^2=0`

`(x-1)^2+(y-3/2)^2-(3/2)^2-1=0`

`(x-1)^2+(y-3/2)^2=13/4`

`(x-1)^2+(y-3/2)^2=(\sqrt13/2)^2`

The above equation is in standard form of circle:

`(x-x_1)^2+(y-y_1)^2=r^2`

Center: `(x_1, y_1)\equiv(1, 3/2)`

Radius: `r=\sqrt13/2`