Question

Convert the polar equation r = 16 sin θ to a rectangular equation.

Answer 1

After converting polar equation r =
`16 sin \theta` to a rectangular equation we get
`x^2 + (y - 8)^2 = 16`. Option C is the right choice.

In rectangular coordinates, x = r cos(θ) and y = r sin(θ). Substituting these into the polar equation r = 16 sin(θ), we get:

x = 16 sin(θ) cos(θ)

y =
`16 sin^2(\theta)`

Using the double-angle identity

sin(2θ) = 2 sin(θ) cos(θ), we can rewrite x as:

x = 8 sin(2θ)

Squaring both equations and adding them, we get:

`x^2 + (y - 8)^2 = 16^2`

Therefore, the answer is:
`x^2 + (y - 8)^2 = 16`

Option C is the right choice.

Answer 2

From the conversion formula, given by

`y=rsin\theta`

we have that

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

By substituting this result into the given polar equation, we have

`r=16*(y)/(r)`

So, by multiplying both sides by r, we get

`r^2=16y`

Now, from the conversion formula

`r^2=x^2+y^2`

we have that

`x^2+y^2=16y`

By subtracting 16y to both sides, we obtain

`x^2+y^2-16y=0`

By completing the square of the quadratic equation on y, we get

`x^2+(y-8)^2-64=0`

then, by adding 64 to both sides, we get

`x^2+(y-8)^2=64`

Therefore, the answer is the last option.