how do you put a polar equation of a conic in terms of x and y?

Question

like r= 12/(6+2sin(t)) is mad into9x^2+8y^2+12y-36=0 according to the book i am confused help pless

Answer 1

Remember the identities.

x = rcost
y = rsint
r² = x² + y²

We have:

r = 12/(6 + 2sint) = 6/(3 + sint)
r(3 + sint) = 6
3r + rsint = 6
3√(x² + y²) + y = 6
3√(x² + y²) = 6 - y

Square both sides.

9x² + 9y² = 36 - 12y + y²
9x² + 8y² + 12y - 36 = 0

Answer 2

The conversion for sin t = y/r and x² + y² = r², so you have

r = 12 / (6 + 2y/r)

r(6 + 2y/r) = 12

6r + 2y = 12

6r = 12 - 2y

divide by 2 and then square both sides:

3r = 6 - y

9r² = 36 - 12y + y²

9(x²+y²) = 36 - 12y + y²

9x² + 9y² = 36 - 12y + y²

9x² + 8y² + 12y = 36
FYI, this is an ellipse