Question

Convert the equation r = 12 sin theta to Cartesian coordinates. Describe the resulting curve.

Choose the correct equation below.

A x^2 -(y+ 6)^2= 36

B x^2+(y? 6)^2 =36

C. (x+ 6)^2+y^2 = 36

D. (x - 6)^2 +y^2 = 36

The equation describes a circle.

The center of the circle is D. (Simplify your answer. Type an ordered pair.)

The radius is g. (Simplify your answer)

Answer 1

Explanation:

Given that there is a polar equation as

`r=12 sin \theta`

This has to be converted into cartesian.

We know the conversion is

`r^2 =x^2+y^2 \\tan \theta = (y)/(x)`

Using this we can say that

`sin^2 \theta = 1-cos^2 \theta \\= 1-(1)/(sec^2 \theta) \\=1-(1)/(1+tan^2 \theta) \\=1-(1)/(1+(y^2)/(x^2) ) \\=1-(x^2)/(x^2+y^2) \\=(y^2)/(y^2+y^2)`

`√(x^2+y^2) =12((y)/(√(x^2+y^2) ) \\x^2+y^2 =12y\\x^2+y^2-12y+36 = 36\\x^2+(y-6)^2 = 6^2`}

Circle with centre (0,6) and radius 6.