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)#
