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)#