Multiply both sides by the denominator:
2r + rsin(theta) = 4
Substitute sqrt(x^2 + y^2) for r and y for rsin(theta):
2sqrt(x^2 + y^2) + y = 4
Subtract y from both sides:
2sqrt(x^2 + y^2) = 4 - y
Square both sides:
4x^2 + 4y^2 = (4 - y)^2
Expand the square on the right:
4x^2 + 4y^2 = 16 - 8y + y^2
Add 8y - y^2 to both sides:
4x^2 + 3y^2 + 8y = 16
Add 3k^2 to both sides:
4x^2 + 3y^2 + 8y + 3k^2= 16 + 3k^2
Change the grouping on the left:
4(x^2) + 3(y^2 + 8/3y + k^2)= 16 + 3k^2
Find the value of k, and k^2 that completes the square in form:
(y - k)^2 = y^2 -2ky + k^2:
y^2 -2ky + k^2 = y^2 + 8/3y + k^2
-2ky = 8/3y
k = -4/3 and k^2 = 16/9
Substitute (x - 0)^2 for x^2 and (y - -4/3)^2 for y^2 + 8/3y + k^2 on the left, 16/9 for k^2 on the right:
4(x - 0)^2 + 3(y - -4/3)^2= 16 + 3(16/9)
Perform the addition on the right:
4(x - 0)^2 + 3(y - -4/3)^2= 64/3
Multiply both sides by 3/64
3/16(x - 0)^2 + 9/64(y - -4/3)^2= 1
Write in the standard form of an ellipse:
(x - 0)^2/(4sqrt(3)/3)^2 + (y - -4/3)^2/(4/3)^2= 1
The center is (0, -4/3)
Force the y term to zero by setting y = -4/3:
(x - 0)^2/(4sqrt(3)/3)^2= 1
(x - 0)^2 = (4sqrt(3)/3)^2
x = +-4sqrt(3)/3
The endpoints of the major axis are (-4sqrt(3)/3, -4/3) and (4sqrt(3)/3, -4/3)
For the x term to zero by setting x = 0:
(y - -4/3)^2/(4/3)^2= 1
(y - -4/3)^2 = (4/3)^2
y - -4/3 = +-4/3
y = -4/3 +-4/3
The minor endpoints are (0, -8/3) and (0, 0)
