Multiply both sides by the denominator:
2r + rsin(theta) = 42r+rsin(θ)=4
Substitute sqrt(x^2 + y^2)√x2+y2 for r and y for rsin(theta)rsin(θ):
2sqrt(x^2 + y^2) + y = 42√x2+y2+y=4
Subtract y from both sides:
2sqrt(x^2 + y^2) = 4 - y2√x2+y2=4−y
Square both sides:
4x^2 + 4y^2 = (4 - y)^24x2+4y2=(4−y)2
Expand the square on the right:
4x^2 + 4y^2 = 16 - 8y + y^24x2+4y2=16−8y+y2
Add 8y - y^28y−y2 to both sides:
4x^2 + 3y^2 + 8y = 164x2+3y2+8y=16
Add 3k^23k2 to both sides:
4x^2 + 3y^2 + 8y + 3k^2= 16 + 3k^24x2+3y2+8y+3k2=16+3k2
Change the grouping on the left:
4(x^2) + 3(y^2 + 8/3y + k^2)= 16 + 3k^24(x2)+3(y2+83y+k2)=16+3k2
Find the value of k, and k^2k2 that completes the square in form:
(y - k)^2 = y^2 -2ky + k^2(y−k)2=y2−2ky+k2:
y^2 -2ky + k^2 = y^2 + 8/3y + k^2y2−2ky+k2=y2+83y+k2
-2ky = 8/3y−2ky=83y
k = -4/3k=−43 and k^2 = 16/9k2=169
Substitute (x - 0)^2(x−0)2 for x^2x2 and (y - -4/3)^2(y−−43)2 for y^2 + 8/3y + k^2y2+83y+k2 on the left, 16/9169 for k^2k2 on the right:
4(x - 0)^2 + 3(y - -4/3)^2= 16 + 3(16/9)4(x−0)2+3(y−−43)2=16+3(169)
Perform the addition on the right:
4(x - 0)^2 + 3(y - -4/3)^2= 64/34(x−0)2+3(y−−43)2=643
Multiply both sides by 3/64364
3/16(x - 0)^2 + 9/64(y - -4/3)^2= 1316(x−0)2+964(y−−43)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(x−0)2(4√33)2+(y−−43)2(43)2=1
The center is (0, -4/3)(0,−43)
Force the y term to zero by setting y = -4/3y=−43:
(x - 0)^2/(4sqrt(3)/3)^2= 1(x−0)2(4√33)2=1
(x - 0)^2 = (4sqrt(3)/3)^2(x−0)2=(4√33)2
x = +-4sqrt(3)/3x=±4√33
The endpoints of the major axis are (-4sqrt(3)/3, -4/3) and (4sqrt(3)/3, -4/3)(−4√33,−43)and(4√33,−43)
For the x term to zero by setting x = 0x=0:
(y - -4/3)^2/(4/3)^2= 1(y−−43)2(43)2=1
(y - -4/3)^2 = (4/3)^2(y−−43)2=(43)2
y - -4/3 = +-4/3y−−43=±43
y = -4/3 +-4/3y=−43±43
The minor endpoints are (0, -8/3) and (0, 0)(0,−83)and(0,0)
